Philosophical transactions. Series A: Mathematical and physical sciences

PHILOSOPHICAL

TRANS AC IONS

OK THE

ROYAL SOCIETY OF LONDON

SERIES A.

CONTAINING 1'APEItS OF A MATHEMATICAL OR PHYSICAL CHARACTER

VOL. 207.

rl

LONDON:

PRINTED BY HARRISON AND SONS, ST. MARTIN'6 LANE, W.C.,

printers in tfrbinanj to %is Jflnjtslg.

FEBRUARY, 1908.

•H

L

v. 2.07

CONTENTS.

(A)
VOL. 207.

List of Illustrations .Mf,0 v

Advertisement

I. The lonisation Produced bij Hot Platinum in Different Gases. By O. W.

RICHARDSON, M.A.. D.Sc., Fellow of Trinity College and Clerk -Max well
Student, Cambridge University. Communicated by Professor J. J. THOMSON,
F-R-S- page 1

II. Second Memoir on the Compositions of Numbers. By Major P. A. MAcMAHON,

R.A.tD.Sc.,F.R.S. 65

III. On the Refractive Indices of Gaseous Potassium, /me, Cadmium, Mercury,

Arsenic, Selenium and Tellurium. By C. CUTHBKRTSON and E. PARR
METCALKK, B.Sc. Communicated by Professor F. T. TROUTON, F.R.S. . 135

IV. On the Discharge of Negative Electricity from Hot Calcium and from Lime.

By FRANK HORTON, D.Sc., B.A., Fellow of St. John's College, and Cl<>rk-
Mti.riri-11 Student of tht Cfotwrnfy, Oambridge. Oammnuucated /•</ /',../:.>,„•
J. J. THOMSON, F.R.S. , 149

V. The Grtirit'itional Stability of the Earth. By A. E. H. LOVE, F.R.S., Sedleian

Professor of Natural Philosophy in the University of Oxford . . . . 171

VI. Investigation "of the Latv of Burning of Modified Cordite. By Major J. H.

MANSKLI., Royal Artillery. Communicated by Sir A. NOBLE, F.R.S. . 243

a 2

VII. On the Dispersion in Artificial Double Refraction. By L. N. G. FILON, M.A.,

D.Sc., Fellow and Lecturer in Mathematics of University College, London.
Communicated by Professor F. T. TROUTON, F.R.S. page 263

VIII. The Distribution of Blue- Violet Light in the Solar Corona on August 30, 1005,
as derived from Photographs taken at Kalaa-es-Senam, Tunisia. By
L. BECKEK, Ph.D., Regius Professor of Astronomy in the University of
Glasgow. Communicated by the JOINT PERMANENT ECLIPSE COMMITTEE. 307

IX. On the Surf ace- Tension of Liquids Investigated by the Method of Jet Vibration.

By P. O. PEDERSEN. Communicated by Lord RAYLEIGH, O.M., Pres.R.S. 341

X. The Normal Weston Cadmium Cell. By F. E. SMITH, A.K.C.Sc. (From the

National Physical Laboratory). Communicated by R. T. GLAZEBROOK,
F.R.S. 393

XI. Electric Furnace Reactions under High Gaseous Pressures. By R. S. HUTTON

and J. E. PETAVEL. Communicated by Professor A. SCHUSTER, F.R.S. 421

XII. A New Current Weigher and a Determination of the Electromotive Force of the

Normal Weston Cadmium Cell. By Professor W. E. AYRTON, F.K.S., and
T. MATHER, F.R.S., Central Technical College, London, and F. E. SMITH,
A.R.C.Sc., National Physical Laboratory, Teddington ....'.. 463

XIII. The Silver Voltameter. Part 1.— By F. E. SMITH, A.R.C.Sc., and T. MATHER,
F.R.S. Part II.— By F. E. SMITH, A.R.C.Sc., and T. M. LOWRY, D.Sc.
Communicated by R. T. GLAZEBROOK, F.R.S. (From the National Physical
Laboratory) 545

Index to Volume 601

LIST OF ILLUSTRATIONS.

Plate 1. — Professor L. BECKER on the Distribution of Blue- Violet Light in tlu^ Solar
Corona on August 30, 1905, as derived from Photographs taken at Kalaa-es-
Senam, Tunisia.

Plates 2 to 4. — Mr. P. O. PEDKKSKN on the Surface-Tension of Liquids Investigated
by the Method of Jet Vibration.

Plate 5. — Mr. F. E. SMITH on the Normal Weston < 'admiuin Cell.

Plate 6.— Messra R. S. HUTTON and .1. E. PKTAVKI. <>n Electric Furnace Reactions
under High Gaseous Pressures.

Plates 7 and 8.— Professor W. E. AYRTON, Mr. T. MATIIKK, and Mr. F. E. SMITH: A
New Current Weigher and a Determination of the Electromotive Force of the
Normal Weston Cadmium Cell.

Plate 9. — Mr. F. E. SMITH, Mr. T. MATHER, and Dr. T. M. LOWRY on the Silver
Voltameter.

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PHILOSOPHICAL TRANSACTIONS.

I. The Ionisation Produced by Hot Platinum in Different Gases.

By 0. W. RICHARDSON, H.A., D.Sc., Fellow of Trinity College find Clerk Maxwell

Student, Cambridge University.

Communicated by Professor J. J. THOMSON, F.R.8.

Received June 19,— Read June 28, 190G.

I. — § 1. INTRODUCTION.

THK principal objects of this investigation have been to examine the part played by
the surrounding gas in the production of ions by hot metals and to discover, if
possible, the mechanism by which the positive ions originate. In what follows,
previous work on ionisation by hot metals will not be described, except in so far as it
bears directly on the questions investigated, since the historical part of the subject
has l>een fully treated in previous papers by the writer,* and others.

The present communication deals -chiefly with the emission of positive ions from hot
platinum, as earlier work has yielded much more information concerning the negative
ioriisation. In 1901 the writer t showed that a great numl)er of facts in connection
with the negative ionisation from hot metals could be explained by supposing that
the electrons, of which the ions consist, were produced in the metal itself, from which
they escaped by virtue of their kinetic energy. This theory makes the negative
ionisation a function only of the metal surface and its temperature, and therefore
independent of the nature and pressure of .the surrounding gas, except in so
far as this may have the effect of modifying the nature of the metallic surface.
H. A. WILSON} has confirmed this part of the theory by showing that the negative
leak, except when ionisation by collision occurs, has the same value in air, nitrogen,
and water vapour over a wide range of pressures. WILSON also showed, however,
tliat hydrogen u-reatly modifies the negative leak. The experiments in the present
paper set-in to show that the effect of hydrogen is due to some change it produces in
the platinum surface; its abnormal behaviour is probably bound up witli its electro-
positive character.

* ' Jahrbueh der Radioaktivitat u. Elektronik.'
t «Crtinl>. Phil. Prcx-.,' vol. 11, p. 286.
J ' Phil. Trans.,' A, vol. 202, p. 243.
VOL. CCV1I. — A 413. B 22.11.06

MR. 0. W. RICHARDSON ON THE IQNISATtON

All the known evidence relating to the ionisation from hot solids goes to prove that
the positive and negative ionisations are, in the majority of cases, entirely separate
effects. It is true that, generally speaking, a hot metal produces ions of hoth signs
simultaneously, but by suitably altering the conditions the ratio of the two ionisations
can be made to change to almost any required extent, even at constant temperature,
and the one can l>e made to vary greatly whilst the other remains practically constant,
so that it is evident that the two ionisations are produced by the operation of at least
two independent causes. The leak from hot solids is, therefore, essentially unipolar.

In the 'Philosophical Magazine' (6), vol. VI., p. 80, the writer showed that the
current from a fresh positively-charged hot platinum wire fell off asymptotically with
the time the wire was heated at constant temperature. An effect of this kind was
first recorded by ELSTER and GEITEL,* who noticed that an insulated plate near an
incandescent platinum wire received a large positive charge when the wire was new,
which gradually diminished, and ultimately changed sign with continued heating.
This initial positive ionisation is presumably independent of the pressure, since it is
very large in a good vacuum. Despite numerous experiments t its origin is still
uncertain, but it is possibly due to some gas or volatile impurity present in the purest
obtainable platinum.

By using a wire in which the initial positive leak had been reduced to a small
value by heating in a good vacuum at a constant temperature at intervals extending
over a long period of time, and subsequently letting in fresh air, the writer} succeeded
in clearly showing that the leak, which was almost independent of the time, consisted
of two parts, one proportional to, and the other independent of, the pressure of the
air. So far as the writer is aware, this is the first experiment recorded which shows
that any part of the positive leak from hot metals is a function of the pressure of the
surrounding gas. The reasons which made such an effect difficult to detect are
(1) the masking of the effect by the initial positive leak, if this has not been
completely removed, and (2) irregularities and time effects in the part of the leak
produced by the gas itself. These will be discussed at some length later in the
paper.

In the present investigation a much more detailed study has been made of the
lonisation in oxygen than in the other gases considered, for several reasons. In the
first place, oxygen is a simple elementary gas which is easily prepared in a state of
considerable purity. It has the additional advantage that small quantities of it
produce a large increase in the positive ionisation which is readilv measured. Finally,
it acts as a self-purifying agent by oxidising, and so getting rid of, hydrogen— an
impurity winch it is of the utmost importance to avoid in experimenting on the
ionisation produced by hot bodies.

* 'Winl. Ann..' vol. :!7, p. 315 (1889).

t 0. \V. Kir iivKi.sr.x, 't'.It. C'nii-iT- Li.^f,' 1905, },. 50

J 'Camli. Phil. Proc.,' vol. 13, p. 58 (1905).

n;oi>rn:i> r.v HOT PI.ATINTM IN DIFKKKKXT CASKS. «

Besides oxygen, the present paper contains an accotmt of measurements of the
ionisation of Ix-tli signs from liot platinum in air, nitrogen, helium, and hydrogen.
There are also measurements of the ionisation from a platinum surface in air when a
calculable quantity of hydrogen is diffusing out from the interior of the platinum.
The last-named experiments shed a considerable amount of light on the mechanism of
the processes by which the ions are produced.

It is necessary to say a word about the use of the term ionisation in this paper.
By "the positive (or negative) ionisation" in a quantitative sense is meant the
number of positive (or negative) ions lilx-ratcd by 1 sq. centim. of the platinum
surface per second under the conditions specified. The ionisations are, therefore,
proportional to the respective saturation currents calculated per square centimetre of
surface.

For convenience of reference the paper has been subdivided as follows :—

I. — § 1. Introduction.

II. — § 2. Experimental arrangements.
III. — The ionisation in oxygen : —

§ 3. Current and electromotive force.

§ 4. Hysteretic relations between current and E.M.F.

§ 5. Current and pressure.

§ 6. Current and temperature.

§ 7. Uncontrollable variationa

§ 8. Comparison of different wires.

§ 9. Special properties of new wires.

§ 10. Theory of the steady positive leak in oxygen.

IV. — § 11. The ionisation in nitrogen.

V. — § 12. The ionisation in air.

VI. — § 13. The ionisation in helium.

VII. — § 14. The ionisation in hydrogen.

VIII. — § 15. Experiments with a platinum tube.

IX. — § 16. Theoretical considerations.

X. — § 17. Summary of principal results.

II. — § 2. E-NPKRIMENTAL ARRAN<;KMKNT8.

K \rept where the contrary is distinctly stated, the arrangement of apparatus used
was similar to that in the author's previous papers.* The platinum wires were
supplied by Messrs. JOHNSON MATTHKY AND Co., and were of the purest material
obtainable. They were O'Ol centim. in diameter and were in the form of a loop, the
wire being about 7 centims. long. The ends of the loop were welded on to stouter
platinum leads (A, fig. 1) which were sealed into one end of a glass tube about

* Cf. ' Phil. Trans.,' A, vol. 202, p. 243.

i: '-'

Mlf. O. W. RICHARDSON ON THE ION1SATION

To CLCCTKOMCTCK

8 centims. in length. The wire was heated electrically as before, and in measuring its
temperature the same Wheatstone's bridge method was made use of to determine its
resistance. This method is an exceedingly sensitive one, and there is no difficulty in
keeping the resistance constant to one or two degrees at the highest temperatures.
This is a matter of considerable importance in working with hot wires when the leak
is a rapidly variable function of the temperature.

The whole of the apparatus, which was in electrical connection with the leads A,
was insulated on paraffin blocks and could be charged to any potential between 0 and
± 800 volts. Owing to the heating current through the platinum filament there was
a fall of potential along the wire amounting to from about 2 to 6 volts according to
the temperature, which had to be taken into account at low voltages. The current
from the wire was measured by means of a Dolezalek electrometer with a suitable

capacity attached to the quadrants. For insulating
the plate B the dry glass inside the tube was found to
be good enough, but outside it was protected by
sealing wax surrounded by a guard ring.

In work of this kind the cleanliness and purity of
the materials employed are of the utmost importance.
In the apparatus shown in fig. 1 it will be noticed that
all the parts are of platinum and glass. This enabled
the tube to be cleaned with boiling nitric acid and
distilled water before the experiments commenced.
The ground-glass joint E enabled a further refinement
to be effected by obviating the necessity of finally
fusing the side tube on to the pump connections and
thereby bringing the cleaned wire into contact with
the gases from the blowpipe flame. The ground joint
was lubricated with carefully purified graphite and was sealed with mercury
externally.

The oxygen used was prepared in two ways. When small quantities only were
required it was obtained by heating potassium permanganate in a tube sealed on to
the apparatus. It was found advisable to have the tube containing the per-
manganate shut off from the rest of the apparatus by a mercury trap as it gave
off a small quantity of gas or vapour even when the salt had been dried by heating
to 120° C. for two hours before it was sealed up. The oxygen produced in this way
is freed from dust by plugging up the front end of the permanganate tube with glass
wool. When larger amounts of oxygen were required the above method was found
to be inconvenient, and the electrolysis of concentrated caustic potash was substituted
for it. The oxygen evolved underwent a preliminary drying by passing through
a tube packed with solid potassium hydrate before being admitted, through a glass
tap, to the main apparatus, where it was subjected to the further action of phosphorus

To PUMP ETC.

Fig. 1.

IM;n|.rrKI>

I'l.ATINTM IN WFFKKENT

peutoxide. As thus prepared, the gas was liable to contain traces ot hydrogen ami
hydrocarbons, but that these were not in sufficient amount to affect the results was
proved by the fact that the oxygen prepared in this way gave the same results as
that obtained from potassium permanganate, which must have been free from
these impurities.

The resistance readings were reduced to platinum temperatures by the method
described by CALLENDAR.* The wire was standardised by determining the melting
point of potassium sulphate by the method previously described, t For the parabolic
correction, which is small at all the temperatures considered, WiLSON'sJ value of the
constant (A = 151) was assumed. This assumption seems justifiable as the value
referred to the same kind of platinum wire.

It has been pointed out above that it is necessary to get rid of the initial positive
leak from hot platinum before experiments can be made on the part of the leak which
is due to oxygen. The following figures give the actual magnitude of the two leaks
for the wire which was employed in this investigation. The wire was cleaned by
boiling with nitric acid and distilled water before commencing and the apparatus was
pumped out to a pressure of 0'00005 inillim. The initial positive leak under these
conditions at a temperature of 804° C. was found to be equal to 1*62 x 1 0~8 ampere.
With the wire at a constant temperature this fell to half in about 10 minutes and,
the rate of decay falling off with time, reached about one-tenth its original value
after an hour's heating. Even after heating the wire for several hours a day for
nearly a fortnight the part of the leak which was independent of the pressure could
not be neglected in comparison with that which depended on the pressure, as the
following numbers, which were obtained at a temperature of 721° C., testify : —

Pressure.

millim.

0-045

0-03

0-016

0-004

0-0003

Current.

ampire
1-8 xlO-"
1-52x10-"
1-4 xlO-'-
9-8 xlO-w
9-6 xlO-"

These numbers show that increasing the pressure from 0 to 0'045 millim. increases
the leak from 9'6 x 10~1S to 1'8 x 10~". The part of the leak due to the gas, for very
small pressures like the above, is very nearly proportional to the gas pressure. The
residual initial leak (9'6 x 10~13 ampere) exhibited by these results was about equal to
the leak produced in oxygen at a pressure of 0'05 millim. ; it was not permanent,
however, but fell away till it could no longer be detected on the electrometer at this

* 'Phil. Mug.' [5], vol. 48, p. 519.
t ' Phil. Trans.,' A, vol. 201, p. 497.
J 'Phil. Trans.,' A, vol. 202, p. 243.

6 MR. O. W. RICHARDSON ON THE IONISATION

temperature. In respect of falling away with time, the initial leak offers a very
marked contrast to the part of the leak which depends on the surrounding oxygen.
This was found to remain constant, except for a temporary variability, under the same
conditions during the whole of the time the experiments were being carried out.
The experiments on this particular wire lasted about three months and during that
time it was heated at various high temperatures for about 150 hours.

In the sequel it will be shown that as the pressure of the oxygen is increased the
current from the positively-charged wire asymptotically approaches a maximum
vulue. It is interesting to compare this value, which is independent of the pressure,
with the initial leak at the same temperature. The experiments show that at 804° C.
the greatest positive leak from an old platinum wire of the above dimensions in
oxygen = 3'6xlO~n ampere, and is therefore about one five-hundredth part of the
initial leak from a new wire.

Another source of trouble in these experiments arises from variations which take
place in the leak when all the controllable conditions are kept constant. These
variations, which will be considered more fully in the sequel, appear to fall into two
classes. The first are of a hysteretic nature and depend on the previous treatment
of the wire. For instance, if the gas pressure is suddenly lowered the leak does not
decrease immediately, but only gradually settles down to its final steady value.
Increasing the pressure gives rise to the converse effect. Sudden changes of
temperature, and in some cases of potential, will be shown to give rise to similar
hysteretic effects. These changes can be explained by supposing that the leak is due
not to the external gas, but to oxygen, which is held chemically or otherwise in the
superficial layers of the platinum, and that the amount necessary for equilibrium
takes time to adjust itself.

The second kind of variation seemed to be of a purely irregular nature, and
manifested itself by sudden jumps in the rate of movement of the electrometer spot
across the scale. This effect was specially marked at high pressures, and at low
pressures was not so noticeable. It may be due to the pressure of the contained
oxygen becoming great enough to force a way through the overlying layer of
platinum, and so giving rise to a sudden evolution of highly ionised gas.

To eliminate errors due to effects of the first kind, some time was always allowed
to elapse after each change had been made, and readings taken from time to time
until the leak became steady. At high pressures, where irregularities of the second
kind occurred in addition, it was more difficult to obtain the equilibrium value of the
leak ; but by neglecting all readings where the movement of the spot was noticed to
be jerky, by always taking the smallest values of the leak, and by waiting till two or
more of these were identical, consistent results could always be obtained. This
procedure was rather tedious in some cases, but it seemed to be the only method, as
the irregularities concerned were not affected by any change in the controllable
conditions (except by diminishing the pressure).

PRODUCED BY HOT IM.ATINTM IN I'HTKIMA'T QA8E8

III. — THE IONISATION IN OXYGEN.
§ 3. Current and Electromotive Fo,

In considering the results of experiments under this head it is important to
ivmrmKtT that owing to the thinness of the wires used most of the fall of potential
occurred near the surface of the wire. For this reason there was always a consider-
ahle electric intensity near the hot wire even when the potential dilll-rence between
the electrodes was quite small. As a sufficient approximation for the experiment-^
described in the next three sections we may take the electric intensity at the surface
of the wire to be 40 V for the wires O'l mi Him. in diameter, and 20 V for the wires
0'2 millini. in diameter, where V is the applied potential difference in volts.

Except within certain limited ranges of pressure and voltage the positive leak from
a hot wire in oxygen was found to be independent of the applied electromotive force.
A saturation current therefore always exists, and it follows that, except in certain
special cases, the number of positive ions produced per second by a hot platinum
surface does not depend on the external electric field. For instance, an experiment
at 700 millims. pressure and a temperature of 793° C. gave the following values
of the current with different E.M.F.'s, the experiments being made in the order
indicated : —

Volts

80

40

200

400

80

200

Current . .

23

23

23

24-5

23

20

0 = 1- 19 xlO-'" ampere)

The values of the current may be regarded as constant within the prolmble
experimental error.

It is evident that at pressures near atmospheric, saturation is attained with a P.D.
of less than 40 volts. As a matter of fact, in these cases, where all the ionisation is
of one sign, there is, of course, no recombination and, provided the whole of the hot
wire is at a positive potential great enough to overcome the tendency to diffuse back
into the wire, Till the ions produced reach the collecting electrode. A saturation
current is thus obtained with a very small voltage. This is shown by some
experiments made at a pressure of the same order of magnitude, 528 millims , as in
tin- last experiment, and at a temperature of 70G° C. The fall of potential along tlie
filament, due to the heating current, was equal to 3'3 volts. The currents obtained
with tin- various assigned -IIK-II, \oltages on the filament are as follows •

Mr.m voltage ....

0

1 75

38

1-75

3-7

1-75 5-8

18

1-75

38

0

Current

4

20

14-8

!- D

15-8

19-5 15-5

1f>

20

14-8

4-8

(l=6x 10*13 umpire)

e

MR. O. W. RICHARDSON ON THE lONISATIo.V

It will be seen that the maximum current is obtained with the potential at the
middle point of the filament equal to +175 volts. Since there was a fall of potential
of 3'3 volts along the filament due to the heating current, the more negative end
would only be at a potential of +0'1 volt when saturation occurred; so that it is
evident that the only competition the electrode experiences in collecting the ions is
that due to the filament itself.

The mean potentials in cases where low voltages were used were determined by
connecting a Weston voltmeter, one terminal of which was earthed, to each end of
the filament in turn. The gradual decrease in the current as the potential rises
from 175 to 38, which is very evident from the above numbers, will be dealt with
later.

These current E.M.F. curves which show saturation at about 2 volts were only
obtained at low temperatures. At higher temperatures the necessary voltage went
up to about 40, as is shown by the following table :—

Mean voltage

+ 0

+ 4-1

+ 16

+ 42

+ 80

-40

Current

5

36

57

63

68

0-04

(1 = 2-4x10-" ampere)

*

Temperature = 1180°C.

Pressure = 89 millims.

The greater difficulty experienced in reaching saturation at high temperatures may
possibly be due to the relatively greater magnitude of the negative ionisation which
would make recombination a factor to be reckoned with.

The experiments show that a potential of +80 volts is sufficient to saturate the
current from a platinum wire at all the temperatures used. This is of great
importance in the sequel, where the leak with 80 volts is used to measure the total
number of ions produced by a platinum wire under various conditions.

When the pressure of oxygen in the apparatus was of the order of one millimeter,
the current ceased to be independent of the voltage at high voltages. A careful
series of measurements was therefore made of the way in which the current varied
with the voltage at different pressures. The results are given in the next table and
are also exhibited graphically in fig. 2. The measurements at the lowest pressure
(O'OG millim.) were made at a temperature of 822° C. ; for all the other pressures the
temperature was 808° C. To eliminate time changes, the leak with + 80 volts was
taken as a standard and was measured both before and after each observation with
another voltage. This precaution was not really necessary, as the steady state had
been reached before the measurements commenced, but it served as a useful check.
On account of the method of taking the observations, and also because they do not all
refer to the same temperature, the absolute values of the leaks have not been given,
but for each pressure they are referred to the current with +80 volts as a standard

i:Y I!<>T 1'LATINTM IN DIFFERENT OASES.

wliicli is put fijual to unity. The absolute values at different pressures can be
obtained from the results for the pressure variation which will be given in a later
section.

Current.

1'ivssmv.

Voltage

i

Voltage
= 80.

Voltage
= 120.

Voltage
= 200.

Voltage
= 240.

Voltage
-280.

Voltage
= 320.

Voltage
= 360.

Voltage
= 400.

million.

:t-46

1-1

0-94

1-0

^_^

0-96

__..

1-24

1-05

1-03

0-96

1-17

l-.'7

1-22

1-69

2-5

0-58

1-06

—

1-24

1-29

1-43

1-70

1-97

3 ;,:,

0-186

—

—

1-19

-

1-65

1-87

2- -27

3-8

0-06

0-81

1-01

—

—

1-38

—

1-40

1-68

200 300

Volts

Fig. 2.

400

These results can readily be explained if we suppose that the wire produces at its
surface a constant number of positive ions per second — which, however, depend, as
will be seen later, on the pressure of the gas — and these ions are all collected
unchanged by the electrode at low voltages. When the voltage increases to 200 or
more, these positive ions produce others by collision and an increase in the current is
obtained. The above increase in the current possesses all the features which are
ivijuired by the view that it is due to ionisation by collision. The change produced
by altering the P.D. from one to another assigned value increases as the pressure is
diminished, reaches a maximum and ultimately disappears. Thus the ratio of the
current under a given voltage to that under 80 volts reaches a maximum as the
pressure is diminished.

The difference in the ett'ect of the positive and negative ions from hot metals in

VOL. OCVII. — A. C

10 MR. 0. W. RICHARDSON ON THE IONISATIOX

producing other ions by collisions is readily seen when the above curves are compared
with similar ones given by H. A. WILSON* for the negative leak from hot plat ii nun
in air at different pressures. WILSON used wires of the same thickness and ;m
apparatus of about the same dimensions as the author, so that the two sets of
observations are quite comparable. It will be noticed that with the negative leak a
given P.D. changes the current in a given ratio at a much higher pressure than with
the positive leak. In addition, the pressures for the maximum current with a given
voltage are much lower for the positive than the negative leak. For instance, when
V = 340, the maximum current for the negative leak is somewhere between 12'1
and 0'81 millim., whereas for the positive leak it is at a pressure somewhere near
0'2 millim. Thus for corresponding effects the value of X/p, where X is the electric
intensity and p the pressure, has to be much greater when the positive ions are the
active agents than when the negative ions are. The actual difference between the
effects of the two kinds of ions is diminished owing to the fact that the negative
ions which the original positive ions produce by collisions also act as ionising
agents.

Curves resembling some of the above have been obtained previously by
McCLELLANDt, who also explained the increase in the current produced by the
electromotive force as being due to the positive ions producing others by collisions.

The author has calculated the number of positive ions which reach a surrounding
cylinder when a given number start from a wire of given radius at its centre and
both the positive ions and the negative ions subsequently produced give rise to
others by collision, in the hope that it might be applied to the further elucidation of
the above experimental results. The expression obtained, however, is so complex
that its manipulation, so as to fit the observations, did not seem likely to greatly
advance our knowledge of the subject. There does not, however, so far as one can
see, seem to be anything in these results contrary to the view that the positive ions
from hot metals have about the same power as the positive ions produced by Rontgen
rays in air of producing other ions by collisions, and that the other ions so produced
consist, in any given case, of an ordinary positive ion together with a corpuscle.

§ 4. Hysteretic Relations between Current and Electromotive Force.

The numbers given in Table 4, § 3, exemplify the curious fact that in some cases the
current increased with diminishing potential difference. The case cited was not
found to be an isolated instance, as the following numbers, obtained at a temperature
of 826° C. and a pressure of 0'4 millim., testify : —

' Phil. Trans.,' A, vol. 202, p. 253 (1903).
t 'Camb. Phil. Proc.,' vol. 11, p. 300 (1902).

l'l«>|>rci.l> BY HOT I'LATINTM IX DIFFKHKXT CASES.

11

Mean volts

-0-9

0

+ 0-4

+ 0-6

+ 0-9

+ 3-2

+ 8-2

+ 8-5

+ 40

r

0-9

1-5

2-8

12

13

20-7

15-7

14-8

13-4

Steady current . . . <
(1 =6x10-" ampere)]

—

—

—

13 .

20
21-7
21

15-4

—

—

Mean steady current . .

0-9

1-6

2-8

12

13

20-8

16-5

14-8

13-4

Similar results were obtained at a pressure of 0'0015 millim.. so that this effect
occurs at all pressures.

The numbers quoted do not give the value of the current immediately the potential
was put on, but what it settled down to later. In all these cases it was found that
on raising the potential the current was too big at first, and only settled down to its
steady value after some minutes. Similarly, on changing to a low voltage the
current was too small at first and subsequently increased somewhat. Effects of this
kind were much more marked with wires which had not been heated very much, and
will be considered more fully under § 13. They can obviously be explained if we
assume that the electric field displaces the equilibrium condition in such a way as to
reduce the number of ionising systems.

Effects of this kind could not be detected at higher temperatures, owing to the
greater difficulty of attaining saturation already alluded to. In these cases it was found
that, though no certain increase in the current could lie detected on changing the
potential from 40 to 80 volts, yet it increased by about 60 per cent, with a potential
of 760 volts. This increase was approximately proportional to the voltage. When
it occurred, 600 volts was used to measure the saturation current ; it was not safe to
go up to much higher voltages, owing to the possibility of ionisatiou by collisions
occurring even at high pressures.

§ 5. Current and Pressure.

We come now to what has been regarded throughout as the main object of this
investigation, namely, the relation between the ionisation produced by the wire and
the preasure of the surrounding gas. In interpreting the results, it is important to
remember (1) that the current was always saturated, and (2) that, at low temperatures
at any rate, the negative leak in oxygen was always found to be small compared with
the positive. These two results conjoined prove that volume ionisation of the gas
round the wire is negligible, at any rate at low temperatures. Moreover, there is every
reason to believe that the negative ionisation, which makes itself felt even in oxygen
at higher temperatures, is the ordinary corpuscular emission from hot metals, so that
everything points to the positive ionisation being due to an action which takes place
at the surface of the metal.

c 2

12

MR. O. W. RICHARDSON ON THK IONJSATIOX

In a previous paper, which has already been quoted,* it was shown that the leak
from a hot platinum wire in air at low pressures fell off as the pressure was reduced
from about O'l millim. to 0 in such a way as to indicate that the leak consisted of two
parts, one proportional to the pressure, and one independent of it. A similar
experiment was now made with pure oxygen, except that observations were started
at a higher pressure. The results of this experiment are given in the next table.
The temperature was 816° C., and the potential on the filament +80 volts.

Pressure in millims. . .

0-000008

0-00028

0-0183

0-116

0-207

0-374

0-675

1-26

2-18

Current

2-15

2-15

5-2

9-5

13

15-5

17

20

24

(1 = 6 x 10-" ampere)

The pressure was varied by making one stroke of the pump and then making an
observation, after waiting for everything to become steady, at the reduced pressure
thus obtained. The order in which the observations were taken was thus that of
diminishing pressure, so that any secular change which might be going on would go

entirely in the one direction and might
vitiate the results. In order to test whether
an effect of this sort was coming in, fresh air
was let into the apparatus the next day and
observations again taken over a similar range.
The results are exhibited along with the
previous ones in fig. 3. The points marked
thus x refer to the observations in Table IX.,
whilst those marked thus O refer to those
taken on the following day. The experi-
mental conditions were the same in both
cases.

The lower curve may be compared with
the numbers for air up to O'll millim. in the
paper referred to above.* It will be seen that
at these low pressures the part of the leak

which depends on the gas is roughly proportional to the pressure, agreeing with
what was previously found for air. The part of the leak which is independent of the
pressure, which is clearly marked in fig. 3, is smaller relatively to the rest than in the
previous experiments. It will not be noticed in the later curves, as it became too
small to be noticeable after a few days' more heating.

Although the results given in fig. 3 show that hysteretic effects had been largely

* 'Camb. Phil, Proc.,' vol. 13, p, 58, 1905.

PRODUCED BY HOT I'LATINUM IN DIFFERENT GASES.

IS

eliminated from the experiments which they represent, still an inspection of the actual
experimental numbers shows that such effects did occur. It was often observed, for
instance, that if the wire had l>een heated for a time to a temperature higher than
that at which olxservations were being taken, the value of the leak was too great for
some time afterwards. For instance, in an experiment at 816° C., where the pressure
was l-4 millims. and the steady leak = 30, the unit being 6 x 10~" ampere, after
heating for a few minutes to about 1100° 0. the wire was found to give the following
values of the leak at the times in minutes stated.

Time ....

0

3

7

11

14

25

28

31

Current .

... 90

70

56

44

40

30

31

30

It is evident from the above numbers that it took alxmt 20 minutes for equilibrium
to be established at this temperature, a fact which gives some indication of the
prodigious labour required in taking the observations.

Another hysteretic effect, which is more likely to be a source of trouble in
experiments on the effect of change of pressure, is a time lag of the change in the
leak behind the change in the pressure. In testing for an effect of this kind it is
evident, from the preceding paragraph, that it is necessary to keep the temperature
constant while the pressure change is made. This is very difficult, since with a
constant current through the wire its temperature is a function of the pressure of
the surrounding gas, but by watching the galvanometer spot of the Wheatstone's
bridge circuit, and manipulating the rheostats which control the heating current as
quickly as possible, the temperature can be restrained from varying very much whilst
the pressure change is l>eing made. The following numbers represent the effect at
809° C. of suddenly reducing the pressure from T8 to 0'21 millim., the P.D. being
+ 80 volts. The steady leak at t'8 millims. had the value 18 (1 = T9 x 10~1S ampere) ;
the leak at 0'21 millim. had the values given at the times stated.

."hours ... 12 12

12

12

12

12

1

1

2

Time <:

Lininutes . . 35 38

41

4G

51

56

4

7

42

Current 14-8 13-7

13 10

9-5

8-1

9

7-8

7-3

1

1

In a similar way, on increasing the pressure, keeping the temperature constant, it
was found that the leak at the higher pressure was too small at first and only
gradually rose to its final steady value.

It is evident from what has been said that the lag in the leak behind pressure

14

MR. O. W. RICHARDSON ON THE IONISATION

B

changes must have affected to some extent the results shown in fig. 3. The practically
exact coincidence of the two curves might merely imply that the observations were
taken at about the same rate in the two cases, so that the errors were about the same
in each case. A check on this was, of course, afforded by the fact that values of the
leak were only retained after they ceased to vary with the time ; but partly owing to
the great length of time required for equilibrium to be established, and partly owing
to the invariable presence of irregularities of another kind, which will be considered
later, it was very difficult to be sure that equilibrium had been attained in any
specified case. For this reason it was thought desirable to have some further means
of checking the extent to which these influences affected the measurements.

This was done by gradually increasing the oxygen pressure instead of diminishing
it, as happened by taking readings after successive strokes of the pump. It is
evident that under these conditions the pressure time lag will have the opposite
effect to what it had when the pressure was being reduced, so that the lack of
coincidence between the curves obtained with increasing and diminishing pressure

will give a measure of the extent to which the
lag effect has not been eliminated. What was
required, then, was an arrangement which would
deliver small adjustable quantities of oxygen
into the main apparatus.

To do this, the apparatus shown in fig. 4 was
devised. The principle of the method is to allow
gas to flow into the apparatus through a very
long narrow capillary tube under an adjustable
difference of pressure for varying times. In the
figure this part of the apparatus is shown
together with the arrangement for furnishing
the oxygen. The bulbous tube A contained
pure dry potassium permanganate ; by heating a
small portion of it for variable lengths of time,
any desired quantity of pure oxygen could be
set free. Such a tube may be used time after
time for long periods. B is a glass wool plug to
stop the manganese dioxide dust which is pro-
duced when A is heated. C is a drawn-out glass
tube which can be broken to let down the
vacuum in this part of the apparatus if required. It is convenient to have the
permanganate tube so that it can be shut off from the rest of the apparatus. This
may be done by means of the mercury cut off I). At F is the fine capillary tube,
bent as shown to economise space, through which the gas flowed into the main
apparatus at H. This tube was about 140 centims. long, and was the narrowest

Fig. 4.

PRODUCED BY HOT PLATINUM IN DIFFERENT OASKs.

15

available in the laboratory. By means of the tube E, which, like D, was filled with
mercury, both being connected to separate reservoirs, the volume above the mercury
at E could be regulated, and so any desired variation could be made in the pressure
driving the gas through the tube F. Finally the glass tap E allowed the time
during which the flow took place to be varied in any desired manner. By suitably
\:trving the two controlling factors it was found that the pressure in the main
apparatus could be increased by any amount from O'OOl millim. to several centimetres
at will. This apparatus works very well and will probably be found to be very
convenient for work of this kind. It permits of a much greater range of variation of
pressure than a regulator which it has been found convenient to employ in previous
experiments, which is practically a tube like E sealed directly on to the apparatus.
With this the pressure is regulated by the change of volume produced when the
height of the mercury column is altered.

Using the apparatus just described, a series of measurements of the leak was now
made, with the pressure of the oxygen gradually increasing. The temperature was
826° C. and the potential on the filament = + 40 volts. The numbers obtained were
as follows : —

Pressure in millions. . ... 0*0107

0-029

0-055

0-12

0-294

0-474

1-09

Current 2-9

4-4

6-6

11-1

19-4

23

35

(1 = 6 xlO-'3 ampere)

30

20

13

C
d

It will be observed that these numbers are very similar to those previously obtained
with decreasing pressure, showing that the time effects do not play a very important
part under the specified experimental con-
ilitions. In order to have a more exact
test, two consecutive series of readings
were taken, one with decreasing and the
other with increasing pressure. The re-
sulting observations are plotted in fig. 5.
The temperature was 828° C. and the
potential 4- 38 '2 volts. The points marked
thus x were taken with the pressure
(U'civasmi;. those marked thus Q with the
pressure increasing.

It will be seen that the two sets of
points fall very nearly on the same curve,
but that those with the pressure increasing tend to be lower than -those with the
pressure decreasing, in agreement with the time-lag effects previously described.

The experiments were now pushed to higher pressures, atmospheric pressure being

10

•33

•67 1-0

FVeaaure : mma.

Fig. 6.

t-83

16

MR. 0. W. RICHARDSON ON THE IONISATION

the upper limit. In doing so, great difficulty was experienced owing to the purely
irregular effects mentioned on p. 6. These effects were characterised by sudden
changes in the rate of leak of a purely temporary kind, and were quite different from
the hysteretic effects previously described. Very often in the midst of a reading,
when the electrometer spot was moving quietly across the scale at the usual rate, it
would suddenly give a kick and dart right off without any warning or apparent
cause. These kicks were much more marked at high than at low pressures. At
pressures of about 1 millim. it was rare for a kick to increase the leak in 15 seconds
by 25 per cent., whereas values of the leak equal to six times the minimum have been
recorded at atmospheric pressure. The following numbers, obtained at a pressure of
366 millims. and a temperature of 809° C., furnish an excellent example of this kind
of thing.

Time.

Leak.

Spot movement.

h. m.

10 31

31-7

steady

10 34

41

slight kick

10 37

33

steady

10 40

200

kick

10 43

31-5

•toady

10 46

33

steady

The above represents the greatest value of the kick recorded. Generally speaking,
the value of the leak only oscillated up to twice its minimum value. Further
experiments on the source of these irregularities will be described later (§ 10). As no
change in the arrangements has been found to eliminate them, they were avoided by
always taking the minimum value of the leak, and readings were always taken until
two consecutive minima agreed with one another. For instance, the true value of
the current from the above observations was taken to be 32. This method was found
to give consistent results.

The curves shown in fig. 6 represent the result of an experiment with the
resistance of the wire maintained equal to that which it would have at a pressure of
1 millim. and a temperature of 816° C., the potential on the filament being +40 volts.
The unit of current is 6 x 10~13 ampere. All the observations except two were taken
with the pressure decreasing. The lower curve represents the same observations as
the upper, but the pressure scale is magnified ten-fold. The two observations marked
thus (*) are on the scale of the upper curve. They were made with the pressure
increasing after the other series were finished, and were taken very rapidly, so that
equilibrium was not fully established. They both fall well below the rest, owing to
tin- time-lag effect described alx>ve. The other points fall very nearly on the curves
drawn.

PRODUCED BY HOT PLATINUM IN DIFFERENT CASKS.

17

400
Pressure

Fig. 6.

The independence of pressure exhibited by the leak at high pressures seemed at
first sight a very surprising result and led the author to enquire whether there might
not be something wrong with the temperature measurements at high pressures. As
has been explained, the criterion employed to obtain a constant temperature so far has
l>een a constant resistance of the wire. Now the resistance measures the average
temperature of the wire across its section, whereas what is required in these measure-
ments is a constant surface temperature, so that any change which alters the
temperature gradient from the centre of the wire to the circumference will alter the
surface temperature at constant resistance. Increasing the gas pressure facilitates
the flow of heat from the surface of the wire and must therefore increase the internal
radial temperature gradient. It is evident, then, that increasing the gas pressure
lowers the surface temperature when the resistance is kept constant. It might be
thought that this effect would be small in the thin wires used (O'Ol centim. diameter),
but tlif leak is a very rapidly varying function of the temperature, so a small
temperature error produces a big change in the leak.

To eliminate this error, which only enters into the experiments on the pressure
variation, and then is only important at somewhat high pressures, a method was
devised by which the surface temperature was kept constant. A tube similar to that
shown in fig. 1 was constructed, exhausted, and sealed up. The wire in it was then
heated to a standard temperature by means of a constant current. A portion of this
filament was then compared with a similarly situated portion of that from which the leak
was being measured, and the heating current through the latter was adjusted until the
t \vo appeared to l>e of the same brightness. Both hot wire tubes were shut up in a
black-lined box, and by looking into this through a tube furnished with paper slits
the field of view could l>e limited to those portions of the filaments which it was
desired to compare.

VOL. ccvu. — A. D

18

MR. O. W. KICHAKPSON ON THE IONISATTON

This method was rather rough and very tedious to use. Great care was necessary
to get reliable results with it, and the strain this involved, added to the natural
difficulties of the experiments, rendered the method almost impracticable. However,
a number of series of observations were taken by this method of direct comparison.
The results of one of them are given in the following table :—

Pressure in millims

0-96

3-8

14

58

135

252

537

747

8-0

9-8

10-7

15

17

19-5

24

22

(1 = 1 • 19 xlO-" ampere)

The temperature was about 800° C. and the voltage + 200. Roughly speaking,
these numbers serve to confirm those which were obtained by the resistance method ;
they show that the leak varies very little with the pressure at high pressures. On
the other hand, there is a more rapid variation than that previously found, indicating
that the resistance method did make the temperatures too low at the higher
pressures.

It was found that a better way to make use of this optical method of obtaining a
constant temperature was to determine the change of resistance required to keep the
filaments equally bright when the pressure was varied, and to use the results thus
got to correct the readings for the leak at constant resistance to what they would be
at constant temperature according to the optical criterion. This procedure may
appear pointless at first sight, but it is not. The advantage of it lies in the fact that
it separates the difficulties of the optical regulation process from those which are due
to the vagaries of the leak itself. The leak then was measured with the wire heated
so that its resistance remained constant ; this is done by the purely mechanical
process of keeping a galvanometer spot at the middle of a scale, so that all the
attention of the observer could be devoted to the actual measurement of the leak
itself. Similarly, in finding the way the resistance changed for the same brightness,
all the attention could be devoted to seeing when the two wires were equally bright.
It was far easier to carry out both these operations separately and combine the
results than to do both things at once, and the results obtained were far more
consistent.

Working in this way a curve was obtained giving the resistance at various
pressures corresponding to a constant temperature. A curve was also plotted, from
experiments which will be described later, showing the relation between the leak at
constant pressure and the resistance. On combining these two curves so as to
eliminate the resistance, a third curve was obtained which gave the factor by which
the leak at any pressure had to be multiplied to bring it to the value it would have
at the temperature which the wire was at when the pressure was 1 millim.

l'l;ol)UCKD 1!V HOT I'LATINUM IN DIFFERENT OASES

19

200

400
Pressure :

Fig. 7.

600

800

Treated in this way the numbers plotted in fig. 6 yield those exhibited in fig. 7.
As In-fore, the temperature is 816° C., the voltage +40, and the unit of current
6 x 10~13 ampere, the pressure being ex-
pressed in millimetres.

It will be seen that despite the tempera-
ture correction which has been made the
leak varies very little with the pressure at
high pressures and is probably asymptotic
to a line somewhere about y = 56.

The preceding observations show that
the positive ionisation produced by a hot
platinum wire in oxygen at temperatures
below 900° C. is approximately proportional
to the square root of the pressure at low
pressures. As the pressure is raised, the
rate of increase of leak with pressure gradually diminishes so that the leak tends to
approach a steady value asymptotically at high pressures. A similar result was
found to hold at higher temperatures, with the difference that the rate of increase
was greater at low pressures. At 1180° C. the leak was nearly proportional to the
pressure below 2 millims., and the rate of increase at higher pressures fell off as
before. The numbers supporting this conclusion will be found below in Part IV.

Before leaving this part of the subject it is necessary to consider another source of
error to which the experiments were liable. This was due to the walls of the tulie
getting heated. During the course of the experiments the author tried the effect of
heating the tube in which the measurements were being made by means of a Buusen
burner placed outside it. This was found to produce an enormous increase in the
leak. In one instance, where the pressure before heating the tube was 0'0005 millim.,
heating for a few minutes with a Bunsen burner increased the leak from 2'2 x 10~'
amj>ere to 5 x 10~* ampere, i.e., in a ratio of 1 to 20,000. At the same time the gas
evolved from the walls only sent up the pressure to O'OOl millim. This curious and
interesting effect, which is being further investigated, does not appear to depend on
the stute of cleanliness of the tube, as it showed itself with apparently undiniinished
vigour after the tube had been taken down and boiled out three times with pure
nitric acid and sul>8equently rinsed out seven times with boiling distilled water. The
effect was also obtained in air at atmospheric pressure and in a vacuum produced by
]ii[iiid air and charcoal, where it was impossible for the wire to come in contact with
mercury vapour or vapours given out by phosphorus pentoxide.

Whatever the cause of this effect may turn out to be, it is clear that it might have
completely vitiated the present measurements of the leak in oxygen and other gases.
In fact, tlu> following oonstderationa will show that a small trace of the above effect
might easily simulate the effects investigated. When the gas pressure is very low,

D 2

20

MR. O. \V. RICHARDSON ON THE TONISATION

very little heat is carried from the hot wire to the surrounding electrode and the
walls of the tube, which consequently remain quite cold. As soon, however, as the
pressure of the gas is increased the temperature of the walls increases too, and if the
above effect were coming into play, the increase in the leak due to the heating of the
walls might easily appear to be a steady function of the pressure of the gas in the
tube.

PUMP ETC.

EL.

EARTH

Fig. 8.

It was considered advisable to settle this question definitely by examining the leak
in a vessel in which the only thing which could possibly get hot was the platinum
wire itself. This was done by means of the apparatus shown in fig. 8. The metal
tube A served as the electrode to which the leak was measured and thus replaced
both the glass tube and the platinum electrode in the previous apparatus. This outer
tube was kept cold by means of water placed in the inverted wide-mouthed bottle B.
A heavy brass tube C, permanently connected to earth and insulated from A by
a clean rubber stopper D, served as a guard ring to prevent leakage from the high
potential wire E across the supports to the testing electrode system AB. The guard
tube C had a heavy flange G soldered to it which formed a base for the apparatus
and rested on the bottom of an earthed biscuit tin F. The lower end of the tube C
projected through a hole in the tin. The hot platinum wire was bound to the thick
copper leads H by means of fine copper wire, and the leads were supported and
insulated from the guard tube C by means of the rubber stopper K. Both the hot
wire leads and the guard tube were kept cold by a stream of water flowing through
the composition spiral PLMQ. The part of this which was laced round the leads H
was insulated from them by thin rubber tube. This was found to conduct heat well
enough to keep the leads cold.

PRODUCED MY HOT PLATINUM I\ DIFFKRKNT GASES.

21

This apparatus, in which effects due to heating of the walls were prevented, was
found to give exactly the same kind of results as the earlier experiments. For
instance, the saturation current in oxygen at different pressures was found to have
the following values :—

Pressure in niillinin. . .

2

0-43

0-07

0-014

124

56

u

2-6

Current

64

38

22

13

228

180

156

99

The temperature in this experiment was 1)76° C. and the unit of current
7xlO~13 ampere. The observations were taken in the order of the numbers in
the table.

Several other points were tested with this apparatus, one of which was to see
whether the lag in the leak behind changes the pressure still held. The wire was
giving a minimum leak of 36 divisions under a pressure of 0/64 millim. when the
oxygen was pumped out as rapidly as possible to a pressure of 0*004 millim. The
temperature was then adjusted to its former value, and the following minimum values
of the leak were observed at the times stated, the time being reckoned from the
point at which the temperature first became steady.

5

10

U

20

Current

24

20

15

15

Evidently the wire requires time to adjust itself to the changed conditions, so that
this effect cannot be attributed to anything given off owing to the walls becoming
heated.

The irregular changes in the leak previously noticed seemed to occur in this vessel
to about the same extent as in the others, so they also cannot be ascribed to anything
from the walls of the tube.

Some puzzling effects which have been observed may, however, probably be assigned
to this cause, and it seems advisable to mention them for the benefit of other workers
iu this subject. The writer has several times obtained a large increase of the leak
with the pressure at high pressures, especially with the wire at a high temperature.
This effect has, however, only been found to occur when the whole tube became very
hot and it could be reduced to a small value by simply blowing cold air on to the
outside of the tube. Another effect which probably arises in the same way is an
increase in the leak at a given temperature produced by heating the wire for a short
time to a c«>nsi(lfral)ly higher temperature (see p. 13). Both these effects appear to
lie really due to the walls of the tube becoming heated.

Mil. 0. W. RICHARDSON OX THE IONISATION

Exj>eriments were also made to see if the negative ionisation in oxygen varied
with the pressure of the gas. The temperature was 1100°C. and the saturation
currents with the wire charged positively and negatively respectively at the different
pressures were those shown in the accompanying table.

0-64

0-15

0-12

0-026

+ ve current

87

27

21

9

- ve current

19

22-5

21

21-5

The potentials used were +40 and —7 '5 volts respectively. These were tested and
found to produce saturation. With the potentials employed there was no possibility
of the measurements being vitiated by ionisation by collisions. The results show
that, whilst the oxygen increases the positive leak ten-fold, the negative remains
unchanged within the limits of experimental error. The independence of negative
ionisation and gas pressure, which had been previously observed by MCCLELLAND and
H. A. WILSON, will be found to be of considerable importance later in interpreting
the results on the positive ionisation.

§ 6. Current and Temperature.

The last section forms a fairly complete investigation of the way in which the
positive ionisation from hot platinum in oxygen varies with the pressure, when the
temperature is kept constant, for a considerable range of temperatures. The
phenomenon was next investigated by measuring the ionisation at constant pressure
when the temperature of the wire was varied.

The measurements were made with the glass tube apparatus previously described.
Rough experiments were made at several pressures, but only those at pressures of
1 to 3 millims. have been retained. At very low pressures irregular results were got,
doubtless owing to changes in the composition of the small quantity of gas present in
the apparatus, whilst at pressures comparable with atmospheric it was feared that :m
rrror might creep in owing to the walls of the tube Incoming heated. At pressures
of about 2 millims., however, several wires were tried and found to give consistent
results over a range from about 700° C. to 1250°C. It was found convenient to
measure the negative current at each temperature along with the positive. The
results therefore enable us to compare the positive and negative ionisations from
wires under identical conditions. The ionisation would, no doubt, be proportional to
the area of the hot metal surface if the surface were uniformly heated. As, however,
the wires are colder ;it the ends nwing t<> the heat being conducted away through the
leads, the effects from wires .if different lengths and thicknesses will not be strictly

IMIOnrCED BY HOT PL. \TINT\1 IN KIKFKKKNT (lASKs.

comparable with their superficial an-as. 'I'o allow tin- this. "'•> rentim. was
subtracted in every case from the length of the wire. The area thus reduced will
be referred to as the effective area of the surface of the wire.

A wire 0*1 millim. in diameter, which had been heated in oxygen for a period of
about three months (usually for several hours a day), was found to give the following
values of the positive and negative saturation currents at the temperatures stated :—

Temperature.

+ Current.

- Current.

°C.

amp&r*

• mjXTT

708

1-6x10-'*

—

770

6-7x10-"

_

826

1-5x10-"

_

BM

:i-2x 10-"

1-1x10-"

940

5-8x10-"

6-7 x 10-"

999

1-1x10-"

8-OxlO-'s

1058

3-8 x 10-'°

6-2 x 10"1*

1119

6-4x10 lu

3-2x10-"

1181

1-1 x 10 -••

3-3 xlO'10

1227

1-7x10 »

l-6x!0-»

The pressure of the oxygen in the above experiment was 1'47 millims., and the
effective superficial area of the wire was 0'223 sq. centim. In this experiment the
negative ionisation was measured with 40 volts P.D. This would probably give
rise to ionisation by collisions at the pressure during the experiment, so that the
values for the negative ionisation in the table are probably somewhat greater than the
true saturation values.

In proceeding from a low to a high temperature it was usually found that the leak
at the high temperature was too big at first and subsequently fell to a smaller value.
This effect did not, however, occur in a wire which had been heated in oxygen for a
very long time — for instance, it was not noticed in the above series of olwervations— -
;iiul is probably due to the wire not really having reached a state of equilibrium.

It will be noticed that the value of the positive ionisation increases very rapidly
with the temperature of the wire, though not so rapidly as the negative corpuscular
radiation. The relationship between the two is brought out more clearly when the
results are exhibited graphically as in fig. U. The continuous curves with points thus x
represent the positive, and the dotted curves with points thus o the negative,
ionisation. The scale of current is different for the different curves. For curves (1)
and (4) the unit is 10~13 amp&re, for (2) and (5) 1 = 10'11 ampere, and for (3) and (6)
1 = 10~'" ampere. The two leaks become equal at ataut 1240° C., though the
positive is fin- Ki^er than the negative at low temperatui. -.

Although the negative ionisation increases far more rapidly than the positive, both
ilfpfiul on the temperature in the same general kind of way ; in fact, the positive
ionisation, so far as its temperature relations are concerned, obeys the law originally

24

MR. 0. W. RICHARDSON ON THE IOX1SATION

40

30

20

10

(2)

700

1300

£>

lv*-"

_. I — >« — ^r^ .— &- .l.-o--

800 900 1000 1100 1200

Temperature : Degrees Centigrade.
Fig. 9.

deduced by the author for the negative ionisation, and can be expressed by means of
the formula A0*e~*'", where A and 6 are constants and 0 is the absolute temperature.
The constant b which measures the work done in setting free an ion is, in general,
much smaller in the case of the positive than the negative ionisation. This may be
tested, as the writer has explained in previous papers, by taking logarithms, when,
if L is the current and A' a new constant,

log,0L/0* = A'-6/(2'300).

The value of the logarithm should therefore be a linear function of 1/0. Values of
log10L/0*, where L is the current per unit area of surface for two wires of different
lengths and diameters, have been plotted against 1/6 in the accompanying diagram.

The extent to which the points fall on two straight lines furnishes a test both of
the applicability of the alxrve formula and also of the nature of the agreement between
different wires. The data for the wires tested are (l) diameter = O'Ol centim.,
effective superficial area = 0'223 sq. centim., positive ionisation-points thus x,
negative ionisation-points thus (x); (2) diameter = 0'02 centim., effective area = O'GG
sq. centim., positive ionisation-points thus • , negative ionisation-points thus O.

It will be observed that the points for the positive leak, and also for the negative
leak, fall very nearly on the same straight line for the two wires. This shows not only
that the leak may be expressed by a formula of the above type, but also that the
constants A and b which enter into the formula are the same for both wires. The
tangent of the angle the above lines make with the axis of 1(0 is a measure of the
work done in setting free an ion. This quantity is evidently much less for the
positive than for the negative ions. The value of this work is conveniently expressed
in calories per gramme molecule of ions, a gramme molecule of ions being the amount
which would occupy 22 '4 litres if in the state of gas at 0° C, and 760 millims.

PROI)IHT.1> |:v HOT I'LATIM'M IN DIFFERENT OASI x

25

-9

-10

-11

"6 _

13

-14

-16

-16

\

600 TOO 800 900 1000 1100

Scale of 10* + 6

Fig. 10.

Expressed in this way, the numbers can at once be compared with the heats of various
purely chemical reactions. The best series of experiments in oxygen (at 1'5 millims.
pressure) gave for the work required to set free a corpuscle o»_ = 13'55 x 104 calories.
This is in good agreement with WILSON'S value 13'11 x 10* for the negative leak from
hot platinum in air at a low pressure.* The value of the work required to set free a
positive ion was found to be tu+ = 3'04 x 10* calories. The heats of the most intense
chemical actions involving only one valency range around 5 x 104 calories, so that the
energy required to liberate a positive ion is somewhat smaller, and that required to
set free a negative ion considerably bigger, than the greatest amount of energy set
free in any known chemical reaction.

§ 7. Uncontrollable Variations.

We have seen (p. 16) that the positive leak in oxygen, particularly at high
pressures, continually varies in an erratic manner, even when all the controllable

» ' Phil. Trans.,' A, 362, vol. 202, p. 269.

vol.. i -CVII. —A.

2fi MR O. W. BICHAIv'DSoN ON THK IOXISATION

conditions are apparently kept constant. The following numbers, which refer to a
temperature of about 900° C., illustrate the kind of thing that occurs. The wire,
which was 0'2 millim. in diameter, was maintained at a potential of -4-40 volts, the
pressure of the oxygen being atmospheric. All the conditions were kept constant ;
the unavoidable variation in the temperature of the wire was continuously registered
by the galvanometer spot, and was not sufficient to cause a variation of 5 per cent.
in the value of the leak. Nevertheless, readings at 2-minute intervals gave the
following values for the current :— 146, 180, 178, 228, 158, 170, 150, 246, 166, 324,
198, 174, 198.

Naturally the existence of a variability of this kind makes it very difficult to find
out what is the real effect of changing the controllable conditions, and the author has
spent a great deal of time in trying to get rid of it. In this he has been unsuccessful.
It looks, in fact, as though this variability is in some way or another an inherent
part of the phenomena. In some respects it seems to follow definite laws. It is
more marked at high than at low pressures, and at low than at high temperatures.
The positive leak at temperatures above 1200° C. seemed to show very little variability.
It is not due to trivial variations in the state of the gas in the tube, as the negative
leak, measured under the same conditions as the positive, did not show it. For the
same reason it cannot be due to discharges from points which might form on the
surface of the platinum.

It was found to be present whatever the voltage on the filament. Boiling out the
tube with nitric acid and distilled water left it unaffected. Slightly heating the walls
of the tube with a Bunsen burner did not affect it, although heating the walls more
strongly was found to give the big leak mentioned on p. 19. The big leak thus
produced, on the contrary, was steady and did not vary capriciously with the time.
The variability was not due to vapour given off intermittently from the heated walls
of the tube, as it occurred in undiminished intensity in the tube whose walls, &c.,
were cooled with water. It might have been ascribed to the intermittent escape of
occluded hydrogen, were it not that a wire which had recently been heated in
hydrogen did not show the effect to a greater extent than a wire which had been
heated in an atmosphere of oxygen over a period of about three months.

There seems to be no escape from the conclusion that this effect is caused by some
periodic change in the state of the platinum surface. If the metal was continually
undergoing recrystallisation accompanied by the emission of absorbed gas, equilibrium
might possibly be incapable of ever being attained, and the results might simulate
those which have been observed. It does not seem advisable to speculate further on
this point. The real question from the point of view of this investigation is whether
the selection of the minimum values in the case of the measurements at high
pressures is legitimate or not. They certainly seemed far more definite than the
maximum or the average values, but apart from this and the apparently remote
possibility that the "jumps" are really due to the escape of some substance foreign

MY IK>T PLATmuM IN DIFFERENT OA8B. 27

to the platinum there is no reason for taking one set of values rather than the other.
< >n these grounds the results at low pressures may he considered more reliuhle than
those at high onea

§ 8. Cnini>nnxon of !)<()'<• rent 11 V/v.v.

After the last section it is refreshing to find that the ionisation from hot wires is
capable in some ways of exhibiting a certain amount of constancy. The results given
on pp. 24 ami -J") show that two wires at any rate gave approximately the same amount
of ionisation, both positive and negative, per unit area with the same pressure of oxygen
at all temperatures from 700° C. to 1250° C. In the course of the investigation the
positive ionisation in oxygen, under similar conditions of pressure and temperature,
was measured for four different wires having different linear dimensions. It is
interesting to compare the leaks from these per unit area of surface when reduced,
by means of the results which have been obtained, to some standard pressure and
temperature. The pressure selected is 1'5 millima, and the leaks at this pressure are
given for two temperatures, viz., 770° C. and 880° C. The data will be found in the
following table (see next page).

The wires numbered (1), (2), and (4) were each O'l millim. in diameter, while
No. (3) was 0'2 millim. in diameter. It will be noticed that, although the area of
the wires varied in the ratio of nearly one to four, and the different wires had been
very differently treated, yet the values of the leak per unit area, as shown by the
last two columns, are very nearly the same in every case. The values furnished
by the first wire on May 5 and July 14 are especially interesting, since they show
that, once the steady condition is attained, there is no further falling off due to
continued heating in oxygen at a red heat. This wire was heated for several hours
on most days between the two dates to a temperature varying between 750° C. and
1100°C. As the observations recorded with wire No. 3 near the temperatures of
770° C. and 880° 0. were a little irregular, the values were selected by drawing a
curve like fig. i) and finding the values of the leak at these temperatures from the
curve. In regard to this wire it is only fair to say that it was subsequently heated
strongly in hydrogen, and after that treatment was found to give a much smaller
leak (about one-third to one-fourth) than before. It seems probable, however, that
this is due to a permanent change produced by the hydrogen in the texture of the
metal surface. It has often been observed that the surface of platinum which has
been heated in hydrogen develops a roughened crystalline appearance. This change
does nut appear to be produced by heating in oxygen, at any rate at temperatures
IM-|O\V 1100° C. The alteration could not U> due, to an error in the temperature
brought alxmt by a change in the temperature coefficient of the resistance, as the
negative leak \\.-is not reduced in as great a ratio as the positive. The figures for
the last \\ire are of interest, as they were obtained in the tulie with water-cooled
walks and wore t lit- re fore free from any effect from the walls.

£ 2

MR. 0. W. RICIIAUDSON ON TIIK IONISATION

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D BY FIOT PLATINI'M IN DIFFERENT GASES. 29

§ 9. Special Properties of New Wires.

The writer* has shown that a new platinum wire, even when well cleaned with
boiling nitric acid and distilled water, gives rise to an abnormally high positive
ionisation when heated in a vacuum. This initial ionisation gradually falls off to a
small value, but it is found that a wire which has lost most of this initial effect still
exhibits peculiar properties when the leak from it in an atmosphere of oxygen or air
is examined. These effects, which only seem to disappear after continued heating at
a low pressure, are characterised by great variation of current with electromotive force
together with time changes in the current under constant voltage.

It has been shown that an old wire exhibits little variation of the current with
electromotive force with voltages above 40. Changing the voltage from +40 to +760
never more than doubled the current in the case of an old wire at atmospheric
pressure. In the case of a new wire, however, a change in the voltage generally
altered the current in the ratio of the applied voltage. The current did not, however,
remain steady at the new value. If the voltage had been increased it gradually fell
to a value much nearer that which it had at the lower voltage. The steady value
which the current seemed to be approaching was greater the higher the voltage, and
what may be called the steady increase with voltage was generally greater the newer
the wire. The converse increase in the current subsequent to lowering the voltage
was also sometimes observed, but was much smaller.

The following numbers, which were obtained in air at atmospheric pressure at a
temperature of about 900° C., with a wire which was not very new and therefore did
not show the effect in its most exaggerated form, will illustrate the kind of thing that
occurred. The wire under +40 volts was giving a leak which oscillated from 33 to
44 divisions, the minimum value of 33 divisions being, however, fairly constant. The
voltage was changed to + 360, when readings every successive 3 minutes gave for the
values of the leak 117, 97, 79, 68, 70, 73. On changing the voltage back to +40,
the leaks at 3-minute intervals were 21, 25, 22, 23.

The above experiments were carried out in the glass tube apparatus first described.
This apparatus relies for its insulation inside the tube solely on the surface of the
glass. It seemed possible that queer effects like the above might be obtained on
changing the voltage if the surface of the glass were getting charged up. This might
be especially likely to happen after a new wire had been sealed into the tube, owing
to the surface retaining a film of moisture, since it was necessary to introduce moisture
to clean out the tube.

For these reasons it seemed advisable to test the question with a form of apparatus
which was not liable to these objections. The apparatus used was really designed for
some experiments on the leak from a platinum tube which will be described later (see
fig. 16, p. 54). The hot wire was insulated along the axis of three equal cylindrical

* ' Phil. Mag.' [6], voL 6, p. 80.

30

ML'. O. \V. IIICIIAUDSON ON TIIK IOXISATION

tubes of aluminium. The two outside tubes acted as a guard ring, so that the leak
was only measured from the uniformly heated central part of the wire to the
surrounding middle cylinder. In addition to this, all the insulation was surrounded
by earthed tubes, so that there was no possibility of it becoming charged up by strong
ionisation. The effects previously observed were fully confirmed with this apparatus.
A new wire, not specially cleaned, placed in this apparatus gave with + 40 volts a
leak of 100 divisions which remained constant for 1 hour 40 minutes. On raising the
potential to +760 volts the following values of the current were obtained at the
times stated : —

{hours ....

2

2

2

2

2

3

3

3

3

3

3

4

minutes . . .

44

47

50

53

58

4

9

20

31

38

44

0

Current

3570

1930

950

760

570

485

475

190

115

112

103

103

On returning to +40 volts, the leaks at successive six-minute intervals were
80, 84, 90, 94. The temperature in this experiment was 925° C. A similar experi-
ment with a hot platinum tube 2 millims. in external diameter showed the same
kind of effect, and also an increase in the positive leak, when the tube had been left
negatively charged.

The obvious conclusion from these experiments is that part of the leak from a new
wire is emitted by something which is itself positively charged, and is therefore
removed when the wire is positive, but not when it is negative. Dust given off by
the platinum might be expected to become positively electrified by virtue 'of point
discharge action in the strong field in the neighbourhood of the hot wire whatever
the sign of its electrification when it left the wire. It would thus be retained by a
negatively, but not by a positively-charged wire. On the other hand, there appears
to be no obvious reason either why dust should produce the observed effects or why a
new wire should give off more dust than an old one. Blowing a current of air past
the wire had no marked effect on the positive leak under a high voltage. The general
complexity of these initial phenomena would seem, however, to indicate that there is
more in them than the above simple explanation demands.

What the process of ageing a wire for the purpose of this investigation consists in
is not quite clear. Mere heating alone will not do it. A new wire, which was heated
lor about a week in the air of the laboratory before being tested, was found to behave
like a new wire when the leak was tested. The above experiments would seem to
indicate that tlie presence of a big positive charge on the wire is instrumental in
Bwifitatifig the decay of the initial leak, whilst, other things being e«|iial. the rate of
decay is greater at low than at high pressures. It looks as though the initial leak

RY HOT PLATINPM IN PIKFKKKNT HASRS.

:i hi'\\ \\irr was «llU' to the presence of Sonic ^.is \vl)ich was |x>sit ivdv
\\licn emit led liv (In- \\iiv. Sucli .-i gas might l»c fxpri-trd In dill'iisc avvav from the
wire more readily at a !<>\v than a high pressure and under a high than a low field.

£ 10. Tlit-ori/ of tin' Sti'inlif I'nx'it /')•/• /,<•///• iii n.ri/tjen.

The view which the author has l>een led to as the result of the foregoing experi-
ments is that the positive, as well as the negative, ions are produced in the first
instance inside the metal, or, at any rate, so far within the surface that their rate of
production is in general uninfluenced by the external field. The experiments, in
addition, afford strong support to the view that the positive ions are furnished not hy
the platinum itself, but by the oxygen absorbed or dissolved in the transition layer
l>etween the metal and the surrounding gas. That the positive ions do not arise in
the space outside the free metal surface appears to be distinctly indicated by the
following considerations.

We have seen that the positive and negative ionisations from hot platinum vary
with the temperature according to the same kind of law, but quite independently of
one another. This is exemplified by the fact that at a pressure of T5 millims. of
oxygen the two leaks are equal to one another at about 1230" C., whereas at 700° C.
the saturation current with the wire negative is about 10~7 of its value with the wire
positive. The disparity between the two leaks points strongly to the view that they
are the result of separate actions ; it proves, in addition, that there is no considerable
amount of volume ionisation, but it does not prove that the ions are not formed
outside the wire. For instance, an oxygen molecule might attack a positively-
charged platinum atom, with the result that of the two oxygen atoms present in the
molecule the one carrying the negative charge combined with the platinum, whilst
the other one was set free, forming a positive ion. Of course, the negative ion would
in a sense be free while the interaction was taking place, but it would always be
within a distance from the attracting positive ion comparable with the radius of the
sphere of molecular action, and the interionic attraction would probably be enormous
compared with the greatest external field of force (about 10* volts per centimetre)
which could l>e applied.

There are two main objections to the foregoing view, which otherwise has the
advantage of simplicity to recommend it. One of these is the way in which the leak
lap* U0iind change of pressure. It will be noticed that the above theory would
make the leak proportional to the rate of reaction between the O, and the free platinum.
In order to ••xplain the variation of the leak with pressure under steady pressures, it
is necessary to assume, what is otherwise probable, that the proportion of uncombined

* This conclusion receives strong support from previous experiments by the writer, in which he showed
that the ionising power of a new wire could be transferred to a second wire, previously rendered inactive
by heating, it" this \\riv negatively charged. 'Phil. Mag.,' [6], vol. 6, p. 86.

32 MR. 0. W. RICHARDSON ON THE IONISATION

platinum is smaller at high than at low pressures. Hence, if the pressure were
suddenly changed from a high to a low value, the leak at constant temperature
should he small at first and then gradually rise to a steady value, while sudden I v
increasing the pressure should give rise to the converse effect. As a matter of fact,
the opposite of this is what has been observed in lx)th cases, indicating that the
ionisation is proportional to the " combined " rather than to the " free " platinum, if
it is permissible to use these words in a rather wider sense than that usually imderstood
by the term chemical combination.

The other objection depends on the way in which the steady leak varies with the
pressure at low pressures. The preceding view would make the leak directly
proportional to the pressure at low pressures at all temperatures, whereas the
experiments show that at low temperatures and pressures the leak varies as the
square root of the pressure. This result is also readily explained if we suppose the
ionisation to be proportional to the amount of combined or absorbed oxygen rather
than to the rate of chemical action between the oxygen and the platinum.

For the reasons stated, it seems necessary to seek the cause of the ionisation
within the surface layer of the metal. On the molecular theory a metal, and in fact
any solid, will possess a surface layer in which the molecules are more loosely held
together than in the interior. On account of the well-known tenacity with which
films of gas adhere to solid bodies, it seems reasonable to suppose that this layer will
be capable of being penetrated by the surrounding gas, and that when equilibrium is
established the absorbed gas will have a density which is very great compared with
that of the free gas outside. This comes to assuming that a gas molecule loses
potential energy when it enters the surface layer. To explain the results it is
necessary to assume, in addition, that, for reasons connected with the structure of the
metal, only dissociated oxygen atoms and not the ordinary molecules are capable of
entering the surface layer. This assumption receives some support from the fact
that the hydrogen which diffuses through hot platinum has been shown to be in the
atomic form.* In order to make the problem definite we shall suppose the surface
layer to be only a few atoms thick, and that each platinum atom is capable of
retaining one oxygen atom in its immediate neighbourhood. The problem, then,
reduces to what is virtually the determination of the condition of chemical equilibrium.
Let the maximum concentration of the free platinum, i.e., the concentration when no
gas is present, reckoned as so much per unit area, be = a, and let the concentration
of the platinum which is used up by retaining its sufficient quotum of gas at time
t = x. Then the concentration of the free platinum at time t = u— x. Let the
external gas pressure = P, then the pressure of the dissociated oxygen = j>, where p
is given by the quadratic

p* = k(P-p) (1),

* Cf. WINKELMANN, ' l)RUDE's Ann..' vol. 8, p. 388 ; and RICHARDSON, NICOI, and PARNKLI., ' Phil.
Mag.' [6J, vol. 8, p. 1.

PHODUCKD I!V HOT PLATINUM IN DIFFKRKNT CASKS. 33

k being the dissociation constant of free oxygen. Tlic rate of formation of the Pt, O
systems is evidently proportional to p (a—x\ whilst their rate of decomposition is
proportional to r. The total rate of increase is therefore

A and B l>eing constants. In the steady state djr/dt = 0, and

Apq a/(P)
= ~ '

if we write b = B/A and p = y*(P). Assuming that the positive ionisation is
proportional to the amount of adsorbed oxygen, its variation with the pressure will
lie given by the above expression.

The assumption that the ionisation is proportional to the amount ot adsorbed
oxygen does not commit us to the view that the ions are liberated by the
decomposition of an oxide of platinum, though it is not inconsistent with that view.
The phenomena of electrolysis show that, when molecular disruption is accompanied
by electrical separation, the metal tends to receive a positive, and the non-metallic
element a negative, charge. We might therefore expect the oxygen to be liberated
from hot metals in the form of negative ions, whereas it is found that the oxygen
receives a positive charge. This seems to indicate that the positive ions are not
charged oxygen atoms when they are expelled from the system platinum-oxygen,
which possibly Incomes unstable owing to continued corpuscular bombardment.
However, since we know so little of the material constitution of the positive ions, it
seems unprofitable to dwell any longer on the question of their precise origin.

Returning to the consideration of the formula that has lx?en obtained for the leak
in equation (2), we see that when the amount of dissociation is small p = &'P*. This
will be the case at low temperatures, if the pressure is not too small, so that under
these circumstances the ionisation will vary with the pressure as aP'/(/8 + I1'), a and y3
taing constants. This expression is evidently of the right form, for when P is small
it varies as P', whilst when P is great it becomes independent of P, and this is the
kind of variation with pressure that was found experimentally. The numbers in the
following table, which refer to an experiment at 730° C., furnish a means of examining
the agreement with the formula in more detail. The unit of current is 10~1J ampere.
The constants in the formula are a = 12, /? = 3'9, the pressures being reckoned in
millimetres of mercury.

The agreement between the observed and calculated values of the ionisation is as
close as the experimental results warrant. In general, the observed leak tends to be
smaller than the calculated at the very lowest pressures, owing to the concentration
of the dissociated oxygen beginning to vary with the pressure more rapidly than as

VOL. CX'VH.— A. F

34

Mil. O. W i;K HARDSON ON THE IONISATION

Pressure in millims.
of Hg.

P>.

Calculated leak,
z = aPV(/3 + P).

Observed leak.

0-006

0-078

0-23

0-17

0-025

0-158

0-465

0-46

0-045

0-212

0-62

0-62

0-077

0-278

0-80

0-795

0-085

0-292

0-84

0-84

0-102

0-32

0-91

0-94

0-143

0-378

1-06

1-23

0-26

0-51

1-39

1-58

0-454

0-674

1-77

1-84

0-95

0-97

2-48

2-5

4-3

2-08

4-2

4-5

17-0

4-13

6-18

6-0

66-0

8-1

8-1

8-7

the square root. The above numbers yield, for the maximum current the above wire
was capable of furnishing, at 730° C. the value I2xl0~12 ampere, or, per square
centimetre of surface, 54 x 10~13 ampere.

The following series refer to a somewhat higher temperature (820° C.) and furnish
a more complete test of the way the formula works at higher pressures. In this case
a was taken = 5G x I0~ia ampere and ft = 4'0 (millims. of mercury)*.

Pressure in millims.
of Hg.

P*.

Calculated leak,
* = aPY(j8 + P).

Observed leak.

0-003

0-055

0-75

1-0

0-17

0-41

5-2

5-9

1-5

1-22

13-2

15

3-1

1-76

17-1

17

6-1

2-47

21-3

20-7

10-7

3-27

25

23-5

17

4-12

28-4

26-5

30

5-48

32-3

30

53

7-28

36-5

34

97

9-85

39-5

38

200

14-3

43-7

43

399

20

46-7

49-3

587

24-2

48-3

50-5

766

27-7

49

53-5

The maximum value of the positive ionisation in oxygen at 820° C. appears from
these numbers to be =2'5 x 10~10 ampere per square centimetre. The values of the above
leaks at the lowest pressures are probably somewhat high, as it was found to be
impracticable to wait long enough to be certain that equilibrium had become fully
established.

We have seen that the assumption that the denominator in the expression for the
positive ionisation is of the form /8 + P* is an approximation which will only hold at

PRODUCED BY HOT PLATINUM IN DIFFERENT OASI -

35

low temperatures. The numbers given below show that at 1170° C. the leak is very
closelv iv|nvsi'iiti"l liv an expression of the form «P/(/J+P), which would be the value
given by the theory if most of the gas outside the wire were dissociated. This is
likely enough to be the case at pressures of the order of 1 millim., and at higher
pressures the exact form of f(P) has very little influence on the result. No doubt a
better agreement could have been obtained by putting in the theoretical value for p>
viz.,

*—£k, but this involves introducing a new constant k into the expression
for the ionisation, a refinement which is not warranted by the exactitude of the
experimental results. It appeared to be impossible to test the theory more fully by
pushing the experiments to a higher order of accuracy, on account of the irregularities
which have already been described.

The experimental values of the ionisation at 1170° C. and those calculated from the
formula aP/(/3 + P) are given in the following table, a was taken = 38 x 10~w amplre,
and ft = 4 '8 millims. of mercury : —

Pressure in milliras.
. of Hg.

Calculated leak,
z = «P/08 + P).

Observed leak.

0-14

0-96

1-08

0-30

2-0

2-24

0-39

2-56

2-85

0-62

3-9

4-35

1-27

7-2

7-9

2-06

10-4

11-3

5-3

18-8

19-9

23-5

30-8

31-5

89

36-0

35-9

The greatest value of the positive leak in oxygen at 1170°C. works out at
17 x 10~8 ampere per square centimetre of platinum surface. The smaller amount of
variation with temperature exhibited by the constant ft is a surprising result.
At the two lower temperatures /? = Bi~'A~' in the notation previously employed,
and its constancy with temperature implies that the heat of formation of the
platinum-oxygen systems from atomic oxygen is nearly equal to the heat of
formation of an oxygen molecule from two atoms. At the higher temperature /? is
equal to B/A, and its value in conjunction with the preceding result implies that
B/A is small compared with 4 '8 at the lower temperatures.

According to the present theory, the energy change associated with the action
which gives rise to a positive ion is given exactly by the temperature variation of the
maximum value of the leak and not by the temperature variation of the leak under
a pressure of I '17 millims. as was assumed on p. 25. The preceding results enable us
to correct for tins, hut as the effect of the correction is smaller than the probable
experimental error it has not been proceeded with.

F 2

36 MR. 0. W. RICHARDSON ON THE IONISATION

IV. — § 11. THE IONISATION IN NITROOEN.

The nitrogen uSed in the experiments was prepared from air and therefore
contained the inert gases in the proportion in which they occur to nitrogen in the
atmosphere. So far as was possible without using very elaborate refinements, care
was taken to free it from oxygen, hydrogen, carbon dioxide, dust and organic
impurities. The air of the laboratory was allowed to bubble very slowly through
a wash bottle containing alkaline pyrogallate and then passed over (l) a red-hot tube
about 60 centims. long containing copper and copper oxide, (2) a long tube packed
with cotton wool, and (3) a tube 30 centims. long packed with fused potash and
plugged with glass wool, before it was admitted into the apparatus. Inside the
apparatus it was finally dried over P205 and further purified (1) at low pressures by
exposure to sodium amalgam from a cathode of which a discharge could be passed
from time to time, (2) at high pressures by heating a copper spiral red hot in it.
The copper spiral had previously been Seated in a vacuum to drive off occluded
hydrogen. With regard to the discharge from the sodium amalgam cathode it was
found advisable not to pass it too frequently, as it absorbs nitrogen rather quickly,
but not so quickly as it absorbs oxygen. All the purifying arrangements were
placed close to the testing bulb and between it and the rest of the apparatus so
as to be as efficient as possible. After the nitrogen had been prepared, the copper in
the hot tube was always found to be quite bright except at the front end.

The positive ionisation in nitrogen is small compared with that in oxygen, especially
at low pressures. As we have seen, a small quantity of oxygen has a very big effect
on the positive leak, and it might therefore be supposed that the leaks observed in
nitrogen were due to traces of oxygen which had not been removed by the purifying
proceases. This, however, is rendered improbable by the fact that the ionisation in
nitrogen differs in character in certain important respects from that in oxygen. The
current requires a greater electromotive force to saturate it under similar conditions,
and it also showed time changes, after altering the applied E.M.F., similar to those
observed in a new wire. These changes were never observed in a wire which had
been heated in oxygen for a long time.

On letting in fresh nitrogen the leak was always found to have a much bigger
value than that at which it finally settled down. The rate of decay was very big at
first, but after a time became extremely slow, so that there still seemed to be a slow
decrease after several hours' heating at 900° C. This decrease may have been due to
the gradual absorption of oxygen. No decrease in pressure could be observed, but
this does not prove anything, as the decrease in the pressure required to account for
the observed effects, supposing them due to oxygen, would fall within the probable
experimental error. The measurements of the relation between the current and the
applied E.M.F., &c., refer to the state in which the leak was varying very slowly, it
at all, with the time.

PRODUCED BY HOT PLATINUM IN' DIFFEKKNT GASES.

.'17

The relation between current and E.M.F. at 2 '8 inilliina pressure and 905° C. is
given by the following numbers : —

Volts

+ 0-3

+ 2-2

+ 7

+ 11

+ 40

+ 80

+ 160

+ 240 + 360

-40

1

Current ...

0-19

0-52

0-66

0-77

ro

1-1

1-3

1-33 1-58

0-53

(1 — 3*9 x 10~ls amp&re per sq. centim.)

At 760 millims. and 920° C. the values found were : —

Volts +40 +120

+ 200

+ 360

+ 560

+ 760

Current 9 26

32

40

•

41

42

(1 = 5-3 x 10~18 ampere per sq. centim.)

Compared with oxygen under similar conditions, these numbers, which are shown
graphically in fig. 11, show that the ionisation in nitrogen requires a greater 'applied

IOO 200 3OO 400 SCO 6OO 7WJ

Fig. 11.

E.M.F. to produce saturation. With the above wire, which was 0-02 centim. in
iliumeter, the current in oxygen showed very little increase with voltage after
40 volts. The increase with 360 volts at 2 '8 millims. is due to the occurrence of
ionisation by collisions.

The variation with temperature of the leak in nitrogen at a pressure of 2*8 millims.
\\.ia also examined. On raising the temperature the leak was too big at first and
usually required about 20 minutes to fall to its minimum value. The numbers
obtained for the positive and negative iouisation are given in the accompanying
table :—

38

MR. O. W. RICHARDSON ON THE IONISATION

Saturation current.

Temperature.

+•

-.

827

ampere
3-0 xlO-18

amp&re
4-4 xlO'14

900

1-7 xlO-12

5-8 xlO"18

907

3-8 xlO"12

1-5 xlO'12

984

2-76x10-"

3-52x10-"

1071

9-9 xlO-"

4-71xlO-10

These numbers were found to obey the formula A#*e *>/a*.

The values of the energy changes associated with the liberation of a gram-molecule
of positive and negative ions respectively are found to be o>+ = 7'12x 104 calories and
W- = ll'2x!04 calories. Thus the work required to liberate a positive ion from
platinum in an atmosphere of nitrogen is more than twice the value of the
corresponding quantity in oxygen.

At low pressures the positive saturation current in nitrogen was found at 905° C.
to consist of two parts, one independent of the pressure and the other nearly
proportional to it. This is shown by the following numbers :—

Pressure in millima

0-067

0-14

0-3

0-6

1-3

2-8

Current

2-6

2-6

2-2

3-0

4-3

6-6

(1 = 3-3 x 10"18 ampere per sq. ceutim.)

The current did not continue increasing with the pressure at the above rate up to
atmospheric pressure, but at slightly higher pressures than the above the current
curve began to bend round towards the pressure axis. It did not, however, tend to
become independent of the pressure as in the case of oxygen, but it increased at
a rate proportional to the pressure at high pressures. The variation of saturation
current with pressure up to atmospheric pressure is shown in the lower curve in
fig. 11. The unit of current = 1:05 x 10~12 ampere per square centimetre. The
temperature was 920° C.

The nitrogen used in the last experiment was allowed to remain all night in the
apparatus, whilst both the platinum wire and the copper spiral were heated in it.
During this time the leak with +40 volts decreased considerably, as is shown by the
following numbers : —

Oxygen was certainly absorbed during the above heating by the copper spiral, for
it was afterwards found to be blackened. It is therefore difficult to be certain that
the final value of the ionisation was really due to nitrogen rather than to a small
trace of oxygen which the copper spiral was incapable of removing. Assuming that

n;oi>rn:i> i;v nor IM.ATINTM IN I>II-TI;I;KNT GASKS.

Time .

I'.M.
5.43

I'.M.

6.01

I'.M.

6.40

I'.M.

7.11

P.M.
9.12

A.M.

1 ,

A.M.

1030

I'.M.

;; MM

48

39

34

28

24

14-5

10-5

9

(1 =- 7 x 10-'* ampere per sq. ccntim.)

oxygen and nitrogen act quite independently of each other in producing the positive
ionisation — reasons will be adduced later for showing that this is improbable —the
proportion of oxygen to nitrogen required to account for the whole of the leak in
nitrogen at atmospheric pressure would be about 1 to 10,000. Against this it is
necessary to set the following facts : —

(1) The ions in nitrogen appear to be considerably bigger than those in oxygen, as
shown by the current E.M.F. curvea

(2) The experiments on air indicate that nitrogen does not simply act as a diluent
to oxygen, but exerts a marked retarding effect on its power of producing positive
ionisation, so that a trace of oxygen present would exert a much smaller effect than
that obtained on the basis of the above calculation.

(3) The positive ionisation in nitrogen varies more rapidly with temperature than
that in oxygen. The numbers indicate that the energy necessary to liberate a
positive ion from hot platinum in nitrogen is rather more than twice the value for
oxygen.

On the whole, the balance of evidence seems to be in favour of the view that
nitrogen produces a genuine positive ionisation which is considerably smaller than
that in oxygen at alxmt 900° C.

V. — § 12. THE IONISATION IN AIR.

The air used in these experiments was purified by passing slowly through two
tubes, one packed with cotton wool and the other with fused caustic potash. It was
suKsequently dried over phosphorus pentoxide inside the apparatus. The general
characters of the results in air are similar to those in oxygen, but the current was
found to be more difficult to saturate under otherwise like conditions.

The relation between the current and the electromotive force at atmospheric
pressure is given by the lower curve in fig. 12. The observations to which both the
curves in fig. 12 refer were made with the water-cooled apparatus already described,
The temperature was 976° C., the effective area of the wire 0'170 sq. centitna,
.-iml the unit of current 7xlO~ls ampere. The upper curve gives the value of the
saturation current at different pressures. In this curve the leak at the four highest
pressures \\.is measured with +704 volts, the next with + 5 12 volts, and the two
lowest with +120 volts. This method of gradually decreasing the voltage as the
pressure was ln\\ered, so as to ensure approximate saturation and at the same time

40

MR. O. W. RICHARDSON ON THE IONISATION

avoid the occurrence of ionisation by collision, was frequently adopted in this
investigation.

1501

800

10O

20O

300 400

Volts

Fig. 12.

500

600 7OO 800

The temperature variation of the approximately saturated leak in air at
atmospheric pressure was also examined with the wire 0'2 millim. diameter in the
glass tube apparatus. The numbers obtained are given in the following table. The
effective-area of this wire was 0'66 sq. centim. :—

Current.

Temperature.

V= +560.

V = - 40.

0 C.

ainix'pc

ampfere

812 9-3 xlO-12

—

893 2-2x10-"

3'3xlO-u

900

5-2x10-"

5-3 xlO-14

978

3-3 xlO-10

3-2xlO-18

1064

8-0 xlO-10

4-2xlO-'2

1150

2-OxlO-9

3-8x10-"

1236

6-7x10-'

2-6 xlO-10

These numbers may be compared with the values for the leak in oxygen at different
temperatures given on p. 23. In doing this it is necessary to multiply the latter by
a factor of about 3 '3, which varies slightly with the temperature, to bring the
positive leak to its value at atmospheric pressure. Botli sets of numbers have also to
be divided by the effective areas of the respective wires. When this is done, it will
be seen that the maximum leak in oxygen is greater than that in air at all
temperatures, though the latter increases more rapidly with the temperature. Both
the positive and negative leaks in air obey the formula A0!r~"/2*, the values of the
constants which determine the energy change associated with the formation of one

piionm-:n i:v HOT PT.ATINTM IN DIFFERENT GASES.

41

gram equivalent of each kind of ions being respectively <o+ = 4 '92x10* and
a*- = 8'97xl04 calories. The value of o>_ for oxygen was 13'5xl04 calories; the
lower value obtained in air is possibly due to —40 volts not being sufficient to
produce saturation. Experiments were not made to test this at the time, but some
results obtained by H. A. WILSON* indicate that this might have been the case. The
value of 4 '92 x 10* for <u+ is intermediate between the values 3 '04 x 10* and 7 '12 x 10*
found for oxygen and nitrogen respectively.

Referring to the above table, we find for the maximum current in air the values
2'2xlO~Mand 5-2xlO~u ampere at 893° C. and 900° C. respectively. Taking the
mean of these, 37 x 10~" ampere, to be the true value at 897° C., and dividing by 0'66
to reduce to unit area, we find for the maximum current at 897° C. the value
5'5 x 10~" ampere per square centimetre. The corresponding quantity in oxygen
deduced from fig. 9 (here we have to multiply by 3'3 on account of the low pressure)
is 5'8 x 10~10 ampere per square centimetre, or ten times the corresponding quantity in
air. Again, referring to fig. 12, we see that the maximum leak in air at 976° C. is
4'5 x lO"10 ampere per square centimetre, a number which agrees satisfactorily with the
value 4<95xlO~10 ampere per square centimetre given by the table on p. 40. The
maximum leak in oxygen deduced from fig. 10 is 12'2xlO~w ampere per square
centimetre, nearly three times the value for air. It is interesting to compare with
these numbers what the values for air would be if the nitrogen had no effect. Since
air contains 21 per cent, of oxygen by volume, on this basis the leak in air at
760 millims. would be the same as that in oxygen at 160 millims. Referring to fig. 7,
we see that this should be equal to 0775 of the value in oxygen at 760 millims.
This would give 44xlO~11 and 9'5xlO~10 ampere per square centimetre at the two
temperatures instead of the values 5'5x 10~" and 4'5x 10~10 actually found.

This result, that the positive ionisation in air was smaller than if the nitrogen had
been removed, seemed at first sight very surprising, and further experiments were
made to test the question directly. This was done by letting air and oxygen into
the apparatus immediately after each other and measuring the leak at atmospheric
pressure. The experiments were carried out at 895° C. The values obtained are
given in the subjoined table, that in nitrogen being added for comparison. The
experiments were carried out in the order of the table, proceeding from above
downwards. The unit of current is 7 x 10~1S ampere.

<;.,..

Saturation current.

Nitrogen

•_".'

Air . . .

70

Oxveoii .

336

Air

70

Oxvfiren

244

VOL. CCVil. — A.

* ' Phil. Traiia.,' A, voL 202, p. 266, tig. 5.
O

42

MR. O. W. RICHAIM)S()N <>N THE TOXISATION

The oxygen iu the first experiment was prepared by electrolysing caustic potash,
in the second hy heating potassium permanganate. The difference in the values for
oxygen is prohahly " accidental."

The disparity hetween the two leaks is considerably smaller than that calculated
from the table, but it shows that nitrogen has a distinct inhibiting effect on the
positive ionisation produced by oxygen. It may do this by combiniug with the free
oxygen ions and so reducing their concentration, or by associating with the platinum
and so reducing the amount available for combination with oxygen. The latter view
would give a satisfactory quantitative explanation of the results.

VI. — § 13. THE IONISATION IN HELIUM.

A few experiments were made in this gas chiefly with the object of finding out if
hot platinum would liberate positive ions in an atmosphere in which the possibility of
chemical action in the ordinary sense was excluded. The helium used was obtained
from Messrs. TYREB AND Co., Sterling Chemical Works, Stratford, E. After
admitting it into the apparatus it was dried over phosphorus pentoxide and purified
by means of a discharge from a cathode of sodium potassium alloy. The tube used
for this purpose is shown in fig. 13. After the apparatus had been completely
exhausted, the liquid alloy was admitted through the stop-cock A. Care has to
be exercised in doing this, as the alloy is liable to stick in the
tube, and, when it begins to move again, to do so with such
violence that it is projected into other parts of the apparatus.
The electrical connection to the cathode was made by the
platinum wire B sealed into the glass. The anode C was of
aluminium. The side tubes D, E led to the pentoxide bulb and
testing vessel and to the pump and McLeod gauge respectively.
MEY* has shown that the discharge from a cathode of sodium
potassium alloy given by an induction coil rapidly absorbs all
gases except the inert ones. The discharge used was found to
run down 3 millims. of air to a pressure comparable with that
due to the residual argon in about 20 minutes. The discharge
was allowed to pass for several hours before measurements of the
leak were made. The helium must have been very pure to
start with, as no decrease in pressure was observed with the
McLeod gauge. The spectrum was also examined from time
to time, and no lines belonging to any gas other than helium
were noticed.

There is every reason to believe that the helium used in these
experiments attained a very high degree of purity, and that the leak observed is really
due to helium and not to traces ot'smne ..iher u,is present. The leak resembled that

* 'DuuuE's Ann.,' vol. 11, p. 138.

Fig. 13.

PROIMVKI* I!Y HOT PLATINTM IN 1HKFKKKNT CASKS.

in nitrogen in showing too big a value after tin- wire had l»een allowed to remain
cold and only gradually settling down t<> a steady \alne. It is very difficult to
explain this effect as being due to the absorption of impuritiea, a* it appeared to be
much the same whether the discharge had lx?en run before the leak was tested or
not. It seems far more likely to he due to a change taking place in the platinum.
The following measurement*) at 907° C. of the leak at 2'4 millims. pressure with
+ 11 volts on the filament show the way it varied with time :—

Time ... . . ...

A.M.

10.46

A.M.

10.48

A.M.

11.10

A.M.

11.25

A.M.

11.50

P.M.

1209

P.M.
12 26

P.M.

2 21

P.M.

2 41

Current

314

294

194

156

134

148

124

134

130

(1 =3'3 x 10~14 ampere per sq. ccntitn.)

The values of the steady leak under +11 volts at the same temperature and
different pressures are as follows : -

Pressure in millinis. of Hg

0-07

0-32

2-4

Current

18

54

130

(1 = 3 -3 x 10"14 ampere per «j. centim.)

The positive ionisation in helium at 2 millims. pressure and 907° C. appears to be,
roughly speaking, about three times that in nitrogen and one-fortieth that in oxygen
under like conditions.

The above experiments are not complete enough to enable us to deduce the law
according to which the leak in helium varies with the pressure. The experiments
were not pushed up to high pressures owing to the difficulty of ensuring the purity
of the gas. They are chiefly of interest as showing that hot metals can produce
ionisation in the inert gases when chemical action is out of the question.

VII. — § 14. THE IONJSATION IN HYDROGEN.

A large number of experiments have been made with this gas, but the phenomena
are very complicated, so that it is difficult to be quite certain of the interpretation of
the results. This is largely owing to our ignorance of the absorption and diffusion of
hydrogen in platinum at low pressures. Some of the effects observed seem to be
of considerable interest and importance, and they will be described in order to give
greater completeness to this account of the subject. Discussion of the theoretical
liearing of the results will be avoided except in so far as it is necessary to render the
purpose of the experiments intelligible, and generally speaking the writer is of
opinion that this jwrt of the work ought to be regarded as a preliminary exploration

G 2

44

MR. O. \V. RICHARDSON ON THE IONISATION

of a very complex question. So far as the negative leak is concerned, the ground
has been previously examined, and valuable results obtained by H. A. WILSON.*

The hydrogen used in the experiments was prepared for the most part by the
action of pure zinc on pure hydrochloric acid with a little ferric chloride added to
make the action go. It was passed over solid potash and cotton wool before being
admitted into the apparatus. In some cases, when only a little hydrogen was
required, it was admitted into the apparatus by diffusion through the walls of a
platinum tube heated in a spirit-lamp flame. This method is a very convenient one
for obtaining a small quantity of hydrogen in a high degree of purity. As is well
known, the negative ionisation produced by hot platinum in an atmosphere of hydrogen
is very big, and it was found impossible to use the electrometer in the ordinary way,
with a condenser attached to the quadrants, in order to measure the currents
obtained. These currents, which amounted to as much as 10~4 ampere in some cases,
were measured by taking the steady deflection when the electrometer quadrants were
connected by a high resistance which could be varied from one to one million ohms.
This was a very convenient arrangement, as it enabled the same instrument, by
simply changing a few plugs, to be used for measuring any current from 10"1 to
10~14 ampere.

Interesting effects are observed when a wire, which has previously been heated in
oxygen only is first heated in hydrogen. Such a wire gives rise to an abnormally
high positive ionisation which gradually decays with time, whilst the negative
ionisation increases simultaneously. The kind of thing that occurs is typified by the
numbers in the subjoined table. The wire, which was O'l millim. in diameter and had
an effective area of 0*214 sq. ceiitim., had previously been heated in oxygen at
2 millims. pressure at a temperature of 800° C. Under these conditions it was
giving about 8'5 x 10~13 ampere positive leak, the negative leak being small compared
with this value. The wire was then allowed to cool, the oxygen pumped out, and
hydrogen let in to a pressure of 27 '5 millims. The currents, with potentials of
— 40 volts on the filament, were then measured at different times at a constant
temperature of 800° C., with the result shown by the table.

hour.

Time S

I minutes . .

33

;.,-.

43

44

47

M

M

88

a

6

10

18

18

21

34

37

1ft

20

28

Current, + . . . .
(1 = 10-" ampere)

M

47-8

ao-s

24

20

14

-

-

18-8

10

7-1

7-1

-

— i

10

7'8

2-6

-

2-1

(1 = 10-* unpdre)

It will be noticed that after the wire had been charged negatively, the positive
leak was abnormally high. This may be a spurious effect due to the insulation

* ' Phil. Trans.,' A, vol. 202, pp. 243 et xtq.

PRODUCED BY HOT PLATINUM IN DIFFKl'KNT CASKS.

liv tin- copious negative ionisation. It was impossible to test tin-
question with the apparatus u-nl.

A second wire, 0*2 milliin. in diameter, was tested and found to give results
almost identical with the ;il>ove when it was heated in hydrogen for the first time at
860° 0. This wire was afterwards heated in hydrogen and later in oxygen for several
days, mostly at a temperature of atxmt 1100°C. A long time after the wire had
again got into a steady condition in regard to the ionisation in oxygen the oxygen
was pumped out and hydrogen re-admitted The wire was now heated in hydrogen
at 26 millims. pressure at a temperature of 900° C., when it was found that the above
slow time changes had almost disappeared. The negative leak when first measured
about 10 minutes after first heating the wire was 672xlO~* ampere, whilst the
positive was 4*5 x 10~la ampere. They subsequently rose and fell to 8x 10~* ampere
and 67 x 10~13 ampere respectively. The difference between the two cases seems to
indicate that heating in hydrogen produces a permanent change in the constitution
of the platinum. There are two other facts which support this contention. One
is the permanent reduction of the value of the steady positive leak in oxygen
produced by continued heating in hydrogen, which was mentioned on p. 27. The
other is that the surface of a wire which has been heated for a long time in hydrogen
becomes visibly pitted and cracked. This change does not appear to be produced by
heating, to moderate temperatures at any rate, in oxygen.

Observations were also recorded of the variation in the ionisation when the gas in
which the wire was heated was changed from hydrogen to oxygen. The change from
one gas to the other was carried out in the same way as in the previous case. Oxygen
was admitted to a pressure of T067 millims. and the wire maintained at 900° C.
Under these circumstances the negative leak was found to fall at once to the
small value previously obtained in oxygen. The negative leak when first measured
registered 2xlO"13 ampere per square centimetre, and it was found to possess the
same value 20 minutes later. The positive leak, on the contrary, fell gradually
during 3 hours from 4'8 x 10~" to 8*9 x 10"" ampere, more than half the fall occurring
in the first half hour. This decrease in the iouisation was accompanied by a slight
decrease in the pressure of the oxygen, which fell to 1*026 millims., indicating that
hydrogen had been evolved by the wire, had combined with the oxygen, and that
the water formed had been absorbed by the phosphorus pentoxide. This experiment
indicates that hydrogen diffusing out of a hot platinum wire increases the positive
leak in oxygen, but is without effect on the negative leak. This conclusion will be
more fully established by experiments to be described later.

The last experiment shows that although a considerable amount of hydrogen may
remain in the wire, the addition of a small quantity of oxygen at once reduces the
negative ionisation to a small value. This indicates that the great negative ionisation
produced by hydrogen in platinum is due to some change of a very superficial
character. On the other hand, a wire which has previously only been heated in

4f,

MI:, o. AV. mriiAunsox ON THE TOMSATION

200

150

5 100

10 20 30

VOIDS
Fig. 14.

4O

only appears to attain to the high value of the leak in hydrogen with extreme
slowness when heated in that gas. It semis very difficult to reconcile these state-
ments if the only time effect occurring is the diffusion of hydrogen into and out of
the wire.

The relation between the negative leak in hydrogen and the applied electromotive

force was next examined. It was found that the normal
curve, exhibiting saturation and a definite relation between
current and electromotive force, could only be obtained
under very restricted conditions. The normal relation
referred to is that exhibited by fig. 14. This experiment
was made with the wire 0'2 millim. in diameter and
0'66 sq. centim. effective area; the pressure of hydrogen
was 3 '8 millims. and the temperature 900° 0.

The heating current caused a fall of potential of about
5 volts along the filament ; the potentials given are the potentials of the middle point
of the filament. The unit of current is l'67x!0~10 ampere. The increase in the
current with voltages greater than 20 indicates that ionisation by collisions was
beginning to come in.

On pushing these experiments to higher potentials, it was found that the current
ceased to be a definite function of the applied E.M.F. and varied in a curious way
with the time. The mystery was cleared up when considerably higher potentials
were applied. The experiments were carried out at 1084° C. with a wire O'l millim.
in diameter and an effective area of 0'214 sq. centim. The pressure was 1'77 millims. ;
the unit of current to which the following numbers refer is 10~8 ampere. It was
found that under a high voltage the steady current was smaller than under a low
one. For instance, under 19 volts the wire had been giving a steady negative leak
of about 147 divisions. At a certain instant the voltage was changed to 286, when
the following values of the current were obtained after the intervals of time
stated : —

Time (minutes)

2

3

5

7

10

13

Current .

62

44

37

33

28-5

26

(1 = 10"8 ampere)

(V = - 286)

The voltage was now reduced to 80, when the current was found to remain almost
steady for some time at 7 divisions. It did not, however, stay at this value, but
after a time began to increase, slowly at first, then more rapidly, then more slowly
again, until it finally became steady at about 220 divisions. This great reduction of
the leak by applying a big voltage and subsequent slow increase under a low voltage

N;o|>rrKI> l:V HOT PLATINUM IN DIFFKKKNT GASES.

47

was observed time after time with two different wires under varied conditions ot
temjKjratuiv and pressure. On the other hand, the absolute values of the leaks

linallv "I. tainccl seemed very capricious. The ^em-nil rharactrr of these time changes
is exhibited by tig. 15.

250

200

150

c
O

100

I

V = 80

V=19

V=286

50 100 150 200 250 300

. Time: Minutes

Fig. 15.

In order to make the experiments comparable, the wire was maintained at a
potential of —286 volts for 120 seconds before the readings under any assigned
voltage were commenced. For obvious reasons the voltage on the filament during
any one set of readings was never changed.

The reduction in the negative leak by applying a big voltage does not occur under
the following conditions: — (1) At very low pressures (<0-1 millim.), (2) at high
pressures (200 millims.), (3) when the wire is positively charged. The subsequent
increase in the ionisation occurs if the wire is either uncharged or charged positively
as wt'll as under a low negative potential. The time required for the establishment
ol'the linai equilibrium appears t<, decrease fairly rapidly as the temperature increiises.
Tin1 voltages employed in the previous experiments were never great enough to
produce a luminous discharge, though the effect does occur if a luminous discharge
passes. The reduction in the ionisation is greater if the wire is made the cathode
than it' it is made the anode.

48 MR. O. W. RICHAR1>SON ON THE TON1SATION

All the above facts point to the view that the reduction of the negative leak in
hydrogen produced by the application of a high potential is due to a change in the
surface caused by the bombardment of the surface by the positive ions produced in
the gas by ionising collisions. This view is supported by the fact that after the high
potential had been applied a greater heating current was always required to maintain
the wire at its original temperature. This shows that the amount of heat radiated
from the surface at a given temperature was greater than before, so that the nature
of the surface must have become changed in some way.

Two views which are not mutually exclusive may be taken as to the nature of the
action by which the bombardment of the positive ions, which are really weak canal
rays, reduces the negative leak. They may act either by destroying a layer of
positively-charged hydrogen which helps the corpuscles out of the metal, or they may
merely allow the absorbed hydrogen to escape from the wire. The last suggestion
receives strong support from the recent experiments of SKINNER on the evolution of
hydrogen from metallic cathodes under the influence of the luminous discharge.* On
the other hand, it is difficult to conceive how bombardment by positive ions for two
minutes can allow so much hydrogen to escape from a wire that it takes several
hours for it to diffuse back again. On the whole, the evidence, though inconclusive,
is in favour of the double layer theory.

The writer has examined the effect of changing the temperature on the value of
the steady negative ionisation in hydrogen, and has confirmed WiLSON'st result that
increasing the temperature gives a leak which is too big initially, whilst decreasing
the temperature has the converse effect. The curve showing the recovery with time,
after heating to a high temperature, is similar in form to that obtained after exposure
to a high potential, although the ratio of final to initial value of the leak was smaller
in the cases examined.

So far little has been said about the steady positive ionisation in hydrogen. We
saw on p. 44 that a wire when first heated in hydrogen gives a considerable positive
leak which gradually decays with time. It is an interesting question whether this
decay would go on indefinitely, or if the positive leak has a minimum value depending
in pressure, temperature, &c. The following experiments show that the steady
positive leak due to hydrogen at 3' 8 millims. pressure and a temperature of 900° C. is
very small, even if it exists.

The wire (diameter = 0"2 millim. and effective area 0'67 sq. centim.) was
maintained at a constant temperature of 900° C. in a hydrogen vacuum for about
3£ hours. By pumping from time to time, the steady pressure was kept below
O'OOl millim., though hydrogen was being given off by the wire. The values of the
leak with + 40 volts at various times were as follows : —

* SKINNKK, ' Phys. Rev.,1 vol. 21, p. 1 (1905).

t H. A. WILSON, 'Phil. Trans.,' A, vol. 202, p. 265.

PRODUCED BY HOT PLATINUM IN DIFFERENT OASES.

49

rhours

12

12

1

l

2

3

Time<

1 minutes

2

55

12

26

44

31

Current

u

12

10-5

9-5

6-7

6-0

(1 - 3 -5 x 10-" ampere)

The rate of evolution of hydrogen from the wire decreased considerably during the
above experiment. During the first half hour the pressure in the apparatus rose by
0-0033 millim., and during the last half hour by 0*00016 millim. The volume of the
apparatus was about 2000 cub. centims., that of the wire being 0-0033 cub. ceutim.
The amount of hydrogen left in the wire would probably be comparable with that
given out in the last half hour, so that the pressure of the hydrogen inside the
wire would, according to the above numbers, still be considerable. The current
21 x 10" l3 ampere at the end of the experiment might be due to the residual gas left
in the wire, so that these experiments are not contrary to the view that the positive
leak in hydrogen, such as it is, is due to absorbed gas.

The effect of letting in hydrogen to a pressure of 38 millims. was now tried, and
the leak at 900° C. measured at various times, with the results shown.

3

3

3

3

3

4

4

5

5

6

Time<

:

Lminutes ....

43

46

49

53

56

1

10

25

39

14

Current

108

104

88

76

68

60

46

18

14

15

(1 = 3-5xlO-'3ampere)

•

The big leak obtained on letting in the fresh gas is a somewhat surprising result,
but might possibly be due to impurities which are gradually destroyed or removed.
The point which seems most important is the small value, 5'2x 10~13 ampere, of the
steady leak at this pressure. This was only two and a-half times the value of the
positive leak obtained after the wire had been heated for 3£ hours in a good vacuum.
The value of the positive leak in oxygen at this pressure and temperature would have
been about 10~"' ampere, or nearly twenty times the above number.

Owing to the smallness of the positive leak in hydrogen, together with other
difficulties which arose, few other satisfactory measurements were made on it.

Measurements of the variation with pressure of the negative ionisation from hot
platinum in hydrogen have l>een made by H. A. WILSON.* WILSON'S method
consisted in measuring the leak when hydrogen at successively increasing pressumi

VOL. ccvn. — A.

* ' Phil. Trans.,' A, vol. 202, p. 243.
H

50

MR. 0. W. RICHARDSON ON THE IONISATION

had been admitted to a wire previously oxidised in nitric acid. At a temperature of
1350° C. the ionisation increased rather less rapidly than if it were proportional to
the pressure up to 0 '01 4 millim. The writer* has made experiments to see if the same
kind of results could be obtained by decreasing the pressure from a high initial value.
The first experiments were made at 900° C., and indicated that the leak with
—40 volts consisted of two parts, one proportional to, and the other independent of,
the pressure. The part proportional to the pressure could be accounted for as being
due to ionisation by collisions, so that the nett result was a leak independent of the
pressure. This leak remained constant when the wire was left hot for 2f hours,
although some gas was given off by the wire, the pressure rising from 0'00033 to
G'0017 millim. This result might be reconciled with WILSON'S by supposing that the
gas was retained by the platinum with extreme tenacity, and that the amount evolved
during the 2f hours' heating was merely an insignificant fraction of what remained in
the wire.

To test this supposition, an experiment was carried out at a much higher
temperature (1390°C.), and an attempt was made to estimate the rate of evolution of
hydrogen by the wire from time to time from the increase in the pressure of the
McLeod gauge. Before commencing the experiment the wire had been heated for
some time in hydrogen at a pressure of 1*35 millims., so presumably equilibrium at
this pressure would have been approximately established. The amount of hydrogen
still retained under these conditions appears to be very large. The rate of increase
of pressure per hour after heatings for the time in hours stated is given by the
following numbers : —

Increase of pressure (millims.)
per hour

0-0064

0-0055

0-0052

0-0050

0-0033

0-0014

•

Mean time (hours) ....

0

2

2-2

3

5-5

11

The numbers are only approximate, as the McLeod gauge was not well adapted for
measuring small pressures accurately. The volume of the apparatus (pump, McLeod
gauge, &c.) was of the order 2000 cub. centims., that of the wire being 0'0033 cub.
centim. On the assumption that all the increase of pressure is due to hydrogen
evolved by the wire, the concentration of hydrogen in a platinum wire at 1350° C. in
equilibrium with hydrogen outside at a pressure not greater than 1 millim. (it may
have been considerably less than this) must be of the order of that corresponding to
a pressure of 2 x 104 millims. of mercury. It seems probable that most of the increase
of pressure is really due to hydrogen evolved from the wire and not from the walls of

* It is only fair to state that WILSON descriljes experiments on the effect of reducing the pressure of
the hydrogen, which gave a much greater diminution in the leak than that oliservcd liy the writer (vule
H. A. WILSON, for. a/., p. 266).

I'ltoltUCKD BY HOT IM.ATINTM IN DIFFERENT GASES.

51

the vessel, sin<v tins j>n»tracted increasing of the pressure in a vacuum was uot
observed after a wire had been heated in other gases. Even if the above large
amount of gas has to be got rid of the rate of escape seems very slow ; in fact, the
numbers show thai the law for the rate of diffusion of hydrogen through hot platinum
obtained by RICHARDSON, NICOL and PARNEIL* at pressures greater than 1 millim.
does not hold at low pressures. It is probable that at these pressures it is necessary
to take external dissociation into account (vide loc. cit.).

Whilst the preceding measurements of the rate of evolution of hydrogen were
being recorded, readings of the current with —13 volts on the filament were taken
simultaneously. This value of the jx>tential was used in order to ensure saturation
(see fig. 1 o) and at the same time to avoid the occurrence of ionisation by collisions
as far as possible. The readings were commenced at a pressure of 0'3 millim., and
the current was found to decrease by about 40 per cent, of its value on reducing the
pressure to 0'002 millim. This additional part of the current, which is nearly
proportional to the pressure, may be accounted for by sujjposing it to l>e due to
ionisation by collisions. The wire was then heated for 16£ hours at a low pressure,
during which time gas was given off at the rates indicated by the numbers in the last
table. The values of the leak (1 = 10~* ampere) and the times, reckoned from the
instant at which the apparatus was first pumped down to 0'002 millim. pressure, at
which they were recorded are given in the next table :—

["hours
Time <

0

0

2

2

3

3

5

6

16

16

Iminutes ....

0

15

16

44

6

17

45

0

10

37

Current

30

26

31

29

30

27

26

26

15

12

At a first glance these numbers indicate a continual falling off in the value of the
leak at constant temperature as the gas escaj>es from the wire. The criterion for
constant temperature was the resistance of the wire, and it was found that, owing to
the spluttering of metal which takes place at high temperatures, the resistance
of the wire at 0° C. had increased considerably during the course of the above
experiment. When this was allowed for it was found that the average temperature
<-t' the wire at the close of the above experiment was 1280°C. instead of 1370°CM
its value when the experiment started. The leak at 1280° C. should have been
alwmt three divisions instead of twelve, so that pumping out the gas had
apparently increased the leak. This paradoxical result is probably caused by the fact
that the above method of reckoning over-corrects for the effect of loss by spluttering.
A calculation from the nuintars in the last table but one shows that the wire had lost

* ' Phil. Mag.,' vol. 8, p. 1.

11 •_'

52

MR. O. \V KlfllAKPSOX ON THE IONISATION

about nine-tenths of the hydrogen originally present in it at the end of the
experiment, so that the experiment appears to warrant the conclusion that the
amount of the negative ionisation depends very little on the amount of hydrogen in
the wire. In fact, the hydrogen appears to act by altering the condition of the
surface of the wire, and once this change has taken place it is very little affected by
changes in the amount of hydrogen either outside or inside. The most reasonable
view appears to be to suppose that the positively charged hydrogen atoms form an
electrical double layer, which helps the corpuscles out of the metal.

A few measurements of the variations of the ionisation, both positive and negative,
with the temperature were made in hydrogen. Experiments at pressures of the
order of 1 millim. were found to be particularly difficult to carry out on account of
the length of time required for equilibrium to be established and the difficulty
of being certain that it was established. The following values, with a pressure of
1'90 millims. represent the best series of measurements at this kind of pressure.

Temperature, ° C

860

1017

1181

Current

2-5xlO~n

13x10-"

112 xlQ-'1

(amperes per sq. centim.)
(V = + 40)

Current

4-7 xlO"10

11 x!0-«

(amperes per sq. centim.)
<V= -40)

These numbers for the positive ionisation will be seen to be considerably greater
when the difference of temperature is allowed for than the minimum value given on
p. 49. This indicates that the steady condition had not really been attained when
the measurements were made, although the final reading was never recorded until
the leak appeared to be varying very slowly, if at all, with the time. [Another
possibility, which must be kept in view, is that these inconsistencies are due to some
other undiscovered factor, which is not taken account of.] If we calculate from the
above numbers the energy change associated with the liberation of a gramme
molecule of ions of each sign we find iv+ = 3'58xl04 calories and IV- = 12'0xl04
calories.

A series of measurements was also made at a much higher pressure (22G millims.).
The numbers obtained are given in the table following.

No regular change could be detected in the value of the positive ionisation at the
lowest temperature over a space of half-an-hour. This tends to confirm the con-
clusion from the experiments on p. 49 that there is a positive leak in hydrogen
which is a function of the pressure. At low pressures this is much smaller than the
positive leak in oxygen, but it increases more rapidly with the pressure. It also

I'KODUCEI) BY HOT PLATINUM IN DIFFERENT OASES.

53

Temperature, * C

860

1017

1097

1181

Current

4-1 x 10~"

3-8 x 10~10

Ux 10~9

(amperes per sq. centim.)
(V = +560)

Current

10 x 10~8

12-5 x 10~*

•>.« x IQ-i

(amperes per sq. centira.)
(V- -40)

appears to increase more rapidly with the temperature. The values of the negative
leak are bigger than those obtained at the lower pressures for the same temperature,
the difference being greatest at the lower temperatures. This would seem to indicate
that the small increase in the negative leak with pressure obtained at the lowest
pressure, and which it was suggested might be due to ionisation by collisions, is
really a genuine direct effect of the gas and becomes greatly magnified at high
pressures.

The values of the energy change associated with the liberation of one gramme
molecule of each kind of ions at this pressure are w+ = 57x10* calories and
W- — 5'56xlO* calories. Thus increasing the pressure of the hydrogen appears to
increase the work required for a positive ion to escape from the metal, whereas it
decreases it in the case of the negative ion. This result so far as it refers to the
negative ionisation has previously been obtained by H. A. WILSON.*

VIII. — § 15. EXPERIMENTS WITH A PLATINUM TUBE.

The writer has also made experiments on the change produced in the ionisation at
the. outside surface of a platinum tube in air when hydrogen was allowed to diffuse
from the inside of the tube. A brief abstract of the results obtained has already
been published f ; the present section gives a more detailed account of the experi-
ments. These platinum tube experiments, in the opinion of the writer, settle
decisively a number of questions which have been, or might be, raised with regard
to the origin of the ionisation produced by hot platinum. For instance, H. A.
WILSON \ has suggested that the negative ionisation produced by hot platinum in
;iir is due to traces of occluded hydrogen which are retained by the wire in a very
persistent manner. If this were the case, the small negative leak in air would be
enormously increased by allowing any considerable quantity of hydrogen to diffuse
out of the wire from inside. As a matter of fact, when hydrogen was allowed to
diffuse out of the walls of the tube at a rate corresponding to 2 cub. centime, at

* ' Phil. Trans.,' A, vol. 202, p. 269.
t 'Camb. Phil. Proc.,'vol. 13, p. 192.
\ 'Phil. Trans.,' vol. 202, p. 243.

54

Mi;. O. W. RICHARDSON ON THE IONISATTON

atmospheric pressure per squ.-m- centimetre of surface per minute, not the slighest
change could be detected in the value of the negative leak. This proves indubitably
that the negative ionisation produced by hot platinum in air is not due to traces of
al»orbed hydrogen.

The apparatus used in this part of the investigation is shown in fig. 1C. The

Fig. 16.

platinum tube ABA' was about 15 centims. long, and its internal and external
diameters were 0'05 and O'lO centim. respectively. It was clamped at each end by
the metal supports E, E', and heated by means of a current let in at D, I)'. The
current of hydrogen or air inside the tube was let through by means of the glass
tubes A, A' sealed on at each end of the platinum tube. The temperature of the tube
was measured by means of the thermocouple C, C' of platinum and rhodioplatinum
welded on to the middle point B. The wire was surrounded by three aluminium
cylinders, F, G, and F', to the middle one of which the leak was measured. The
outer cylinders acted as guard rings and maintained a uniform field near the central
uniformly heated part of the tube B. The dimensions of the middle cylinder were :
length = 3 centims., diameter = 3'2 centims. The various supports E, I, 1', E', could
slide in holes cut in a slab of vulcanised fibre LL', which was used because ebonite was
found to buckle with the heat. When they had been adjusted in position they could
be clamped by means of screws. The vulcanised fibre was not found to be sufficiently
good insulating material, so the support L to the testing electrode was protected by

PRODUCED BY HOT PLATINUM IN DIFFERENT OASES. 55

an earth-connected tube J in which it was held by an ebonite cylinder K. The whole
of this part of the apparatus was fixed in a wooden box covered with lead foil
connected to earth.

In reducing the thermocouple readings to temperatures, the platinum temperature
was first calculated by making use of the reading corresponding to the melting-point
of potassium sulphate. This point was determined experimentally in the way already
descril>ed. The platinum temperatures were then reduced to centigrade by means of
the correction curve given by CALLENDAR.* The legitimacy of this process was
tested by making an independent determination of the melting-point of sodium
sulphate. The value found was 885° C., and is within 2° of that (883° C.) given by
HEYCOCK and NEVILLE for this constant.

Sealed on to A' were a mercury manometer, a glass bulb of about 300 cub. centims.
capacity, and a glass tap. The last named was connected to the apparatus for
delivering and purifying the hydrogen which was prepared, as described previously, by
the action of pure zinc on hydrochloric acid. From experiments on the diffusion of
the hydrogen through the walls of the platinum tube it is l)elieved to have been
exceptionally pure. The tube A was also sealed on to a three-way tap so that the
hydrogen or air could be either sucked by means of a water pump or allowed to
bubble through water. These arrangements made it easy (1) to test if the hydrogen
was diffusing through the tube properly, (2) to replace the stream of hydrogen by air
and vici' versd, and (3) to change the pressure inside the apparatus which regulated
the rate of diffusion of the hydrogen through the walls of the tube.

It is convenient to consider first the effect of the hydrogen on the negative
ionisation in air. Preliminary experiments showed that the current could not be
saturated by the voltages at the writer's disposal, so the current with —80 volts was
measured instead of the saturation current. This makes the absolute values of the
currents considerably smaller than those previously obtained, more especially as the
latter are probably greater than the normal on account of ionisation by collisions (see
p. 23). The first test was made at 1200° C. After the tube had been heated for a
long time with air both inside and outside, the current with —80 volts was found to
lie 21 x 10~u ampere per square centimetre of surface. The tube was then allowed to
cool, the air replaced by hydrogen inside, and the leak again measured. With
hydrogen inside at a pressure of 115 millims. the leak under the same conditions was
26xlO~u ampere per square centimetre, and with hydrogen at atmospheric pressure
24xlO~14 ampere per square centimetre. These numbers are all equal within the
proliable accuracy of the temperature regulation.

Another experiment was made at 1380° C. The current with —80 volts with
hydrogen inside the apparatus at atmospheric pressure was found to be 37 x 10~*
ampere per square centimetre, and with the hydrogen at 65 millims. the current was
3'G x 10"" ampere per square centimetre. In the latter case the amount of hydrogen

* 'Phil. Mag.' [5], vol. 48, p. 519.

56 MR. O. W. RICHARDSON ON THE IONISATION

diffusing through the wire would have been rather less than one-third of what it was
in the former. A calculation based on the results of RICHARDSON, NICOL, and
PARNELL* showed that the amount of hydrogen diffusing through each square
centimetre of surface of the platinum per minute must have been equal to about
2 cub. centims. at 0° C. and 760 millims.

Since the platinum tube in these experiments was giving the small negative
ionisations normally produced in air before the hydrogen was allowed to diffuse
through, the above experiments prove indisputably that the negative ionisation
produced by hot platinum in air and other gases is not due to residual traces of
absorbed hydrogen. It appears to be possible to go further than this and to say that
the effect, on the leak, of hydrogen inside the metal is not due to its direct action as
hydrogen, but to some change it produces in the properties of the metal surface.
This change appears to be inhibited when the metal is heated in an atmosphere of air
or oxygen.

To substantiate this conclusion it is necessary to prove that there was enough
hydrogen inside the outer surface of the wire during the experiment to have
appreciably altered the value of the negative leak if it were exerting its full effect.
This may be done by finding a minimum value of the external hydrogen pressure
which would just stop the diffusion outwards for an instant. If the velocities of
the escaped hydrogen molecules were suddenly reversed, the diffusion would stop
momentarily, and the external pressure then occurring would give the minimum
external pressure which would keep the hydrogen inside the surface layer in equi-
librium. The equilibrium pressure might be greater than this, but could not be less.
In the experiment the mass of hydrogen diffusing through 1 sq. centim. per second at
1380° C. = 2'65xlO~7 gramme. In free hydrogen at 1 centim. pressure the mass
which is carried across an area of 1 sq. centim. per second = 2'3xlO~2 gramme at
1380° C. Hence the minimum value of the external pressure with which the
hydrogen instantaneously present inside the surface layer could be in equilibrium is
1*15 x 10~4 millim. It is necessary to show that a pressure of this amount of hydrogen
would have produced an appreciable increase in the value of the leak. According to
one table given by H. A. WILSON (loc. cit., p. 265), at 1350° C. hydrogen at 6x 10~4
millim. pressure increases the negative leak by a factor of 2500. The writer,
however, is inclined to think that a more accurate comparison with the present
experiments can be got by comparing the tables on pp. 260 and 269 of WILSON'S
paper. These show that hydrogen at a pressure of 13xlO~4 millim. increases the
leak at 1375° C. by a factor of 8, so that l'15x 10~4 millim. would cause an increase
by a factor of not less than 1'6. The experiments recorded in the present paper
(p. 52) also indicate an increase in the leak of about 100-fold at 1340° C. due to
hydrogen at a pressure certainly less than 10~a millim. On the assumption that the
negative ionisation is nearly proportional to the pressure, this would give about the

* ' Phil. Mag.' [6], vol. 8, p. 1.

PRODUCED BY HOT PLATINUM IX DIFFERENT OASES. 57

same increase due to 10 ' milliin. as that obtained above. Reasons bave been adduced
i Mi-lier in this paper (p. 52) for believing that the assumption that the negative
iouisation is nearly proportional to the pressure of the hydrogen is incorrect. It has
been retained in the present argument because it is the assumption which is most
unfavourable to the view advocated. On the most unfavourable view, then, the
hydrogen diffusing through the wire should have produced an increase of at least
60 per cent, in the leak. Allowing an exj>erimental uncertainty of 20 per cent., no
change could be detected in the leak due to the diffusion of hydrogen through the
wire. This strongly supports the view, which also seems required on other grounds,
that the hydrogen does not act per se, but produces some change in the platinum
surface, and this change is prevented from taking place if the platinum is heated
in air.*

In contrast to the negative ionisation, the positive ionisation produced by the hot
platinum tube was found to be altered when hydrogen was allowed to diffuse through
from inside. In fact, at constant temperature an additional amount of ionisation is
caused thereby which is proportional to the amount of hydrogen diffusing through
the tube.

At high temperatures the positive ionisation was found to be readily saturated.
Thus at 1200° C. the leak with -|-80 volts was equal to 64 divisions, and with
+ 400 volts 75 divisions with air inside the tube ; with hydrogen diffusing through,
the values under these voltages were 88 and 95 divisions respectively. These
proportions were much the same, so the leak was generally measured with +80 volts,
as higher voltages were not always available. On changing from a low to a high
voltage, a big leak was often noticed at first, but this always fell away in a few
minutes, until, approximately, the above ratio was obtained. Effects of this kind
have already been described in detail (see § 9). At low temperatures the positive
ionisation obtained with this apparatus seemed to be different in character, for it was

[* Note added Septemlrr 7, 1906. — It seems advisable to indicate the exact bearing of this argument
more precisely. It is intended to confirm the conclusion, which has Ixsen drawn from direct experiment
on page 52, that the high value of the negative ionisation in a vacuum containing traces of hydrogen is
due to the fact that the hydrogen keeps the surface of the wire in a certain state, rather than that the
wire contains a certain amount of hydrogen. The writer does not wish to create the impression that
hydrogen never exerts a direct influence on the magnitude of the negative leak. The numbers on
p. 53 show that the value of the ionisation, at constant temperature, increases with the pressure of
the hydrogen at high pressures ; so that it is probable that at high pressures there is a negative leak
which is a definite function of the pressure of the hydrogen. This is also demanded by the fact that the
constant «o_, which enters into the temperature formula, is dependent on the pressure.

A comparison of the table at the top of page 63 with that on page 52 would seem to indicate that the
ionisation in hydrogen at 1'9 millims. pressure is much greater than in a hydrogen vacuum. The writer
considers, however, that these experiments are not comparable with one another, owing to the wires having
lM?en differently treated before the two experiments. The direct testa made on pp. 50 and 51 showed that
decreasing the pressure from about 1 millim. to -001 millim, only reduced the ionisation in hydrogen by
about 40 per cent, of its value.]

VOL. CCVII. — A. I

58

MR. O. W. RICHAEDSON ON THE IONISATION

impossible to saturate it. This is shown by the following numbers for the positive
current under different voltages at 809° C. :—

Volts +

0

4

10

20

40

80

400

960

Current

0

2-6

10

22

32

64

225

390

(1 = 1 -8 x 10~12 ampere per sq. centim.)

A number of experiments were made to find the cause of this anomalous behaviour
at low temperature, but no satisfactory conclusion was arrived at. Fortunately, this
does not matter much as far as the present investigation is concerned, for the
experiments described below were all carried out at much higher temperatures, when
saturation was very nearly attained with 80 volts.

Experiments on the variation of the positive ionisation with the rate at which
hydrogen was diffusing through the walls of the tube were made at 1200° C.
approximately. The rate of diffusion was varied by varying the pressure of the
hydrogen inside the apparatus, since the quantity diffusing in a given time has been
shown* to vary very nearly as the square root of the pressure inside. The way in
which the saturation current varied with the pressure P of the hydrogen inside the
tube is shown by the following numbers : —

Current.

(1 = 1 -8 x 10~n amperes per sq. centim.)

P.

Found.

Calculated.

millims.

0

42

42

30

51

52

60

56-3

55

172

64

65-6

780

90

92-4

The numbers in the last column were calculated by assuming that the current was
equal to a + ftP*, a and b being constants. The agreement of the results shows that
the leak consists of two parts, one independent of, and the other proportional to, the
square root of the pressure of the hydrogen inside the tube. The effect of the
hydrogen diffusing out of the platinum, therefore, is to produce an additional number
of positive ions proportional to the amount of hydrogen diffusing out.

These results tend to indicate that the hydrogen inside the metal, which is known
from other considerations to be in the atomic state, is positively charged. Only a
small fraction (about 10~7) of the hydrogen comes out in the ionic form, but on

* RICHARDSON, NICOL, and PARNELL, loc. cit.

PRODUCED BY HOT PLATINUM IN DIFFERENT GASES.

59

account of the electrostatic attraction the charged atoms might be expected to have
greater difficulty in escaping from the metal. This would especially be the case at
low temperatures, and may account for the hydrogen set free from palladium not
being ionised. These considerations are also in agreement with the fact that
electrolytic hydrogen, which is positively charged, is capable of diffusing into some
metals — for instance iron — at ordinary temperatures.

Experiments were also made to see how the positive ionisation in air varied with
the temperature, (1) when the tube had been heated for a long time in air and there
was no hydrogen inside the tube, (2) with a constant pressure (atmospheric) ot
hydrogen inside. The results, which extend from 973° C. to 1331° C., are exhibited in
fig. 17 ; the numbers in brackets denote the order in which the observations were

700
600
500

,400

I

;300

200

100

m

/"»

950 1050 1150 1250 1350

Temperature: Degrees Centigrade

Fig. 17.

taken. The upper curve represents the ionisation with hydrogen inside the tube, the
lower one that without. The difference between corresponding ordinates represents
the part of the ionisation which is due to the hydrogen diffusing through at any
temperature. The values of the hydrogen part of the current (1 = 1*8 x 10~u ampere
per square centimetre) at various temperatures are given in the following table :—

Temperature, * C

973

1052

1129

1200

1262

1331

Current

6

17

13

80

172

340

These numbers increase much more rapidly with the temperature than the
quantities of hydrogen diffusing as given by RICHARDSON, NICOL, and PARNELL'S
experiments. Hence the efficiency for producing positive ionisation of a given
amount of hydrogen diffusing out of platinum increases rapidly with increasing
temperature.

I 2

C,0 Mil. 0. \V. RICHARDSON ON THE tONISATlON

The fact that hydrogen diffusing out of a platinum wire produces a positive
ionisation proportional to the amount of gas diffusing, taken in conjunction with the
fact that the additional ionisation so produced, even when the amount of gas
diffusing per minute is equal to 1 cub. centim. at 0° C. and 760 millims. per sqiiare
centimetre, is only about equal to the positive leak when no hydrogen is apparently
present, proves that the positive ionisation in oxygen and other gases as well as the
negative is not due to residual traces of absorbed hydrogen. This position has
already been shown to be highly probable by other considerations, the chief of which
are: — (1) The constancy of the positive ionisation in oxygen with long-continued
heating, (2) the agreement between different wires, (3) the fact that heating a wire
in hydrogen seemed to produce a permanent decrease, and not an increase, when the
steady ionisation it produced in oxygen was measured subsequently.

IX. — § 16. SOME THEORETICAL CONSIDERATIONS.

The above experiments show that the steady positive ionisation produced by hot
platinum in different gases, so far as its variation with temperature is concerned,
obeys a formula first deduced by the writer* and shown to represent the negative
corpuscular ionisation from hot platinum. That this would be the case was rendered
highly probable by the fact established by the writerf some time ago that the
temperature relations of the positive ionisation, when it is changing with time, were
adequately expressed by the formula C = A^e"9'2*, A and Q being constants. The
only theoretical conclusion which this temperature relation seems to warrant is that
the liberation of an ion occurs when the dynamical system from which it is produced
acquires a certain amount of energy, which is furnished, it may be indirectly, by the
energy of thermal agitation of surrounding systems. It does not really afford any
evidence as to whether the production of ions is, or is not, accompanied by chemical
action.

It is interesting to compare the values of Q, which represent, on the assumption
that equilibrium is possible, the amount of energy in calories associated with the
production of 1 gramme equivalent of ions. The numbers which were obtained are
given in the following table :—

QM.

Pressure.

Q+.

Q-

N+.

N_.

Oxygen

millims.

2

3-04 x 104

13-55 x 104

5 x 1010

3 x 10-s

Air ....

760

4.00 x 104

s-07 x 104

•"> x 1013

101S

Nitrogen «

2-8

7-12 x 104

H -o x 104

3 x 1016

'2 x 10S1

Hydrogen . ,

1-9

3-58 x 104

12-0 x 104

10"

10-6

Hydrogen

226

5-7 x 104

5 -5G x 104

1015

2 x 1014

* 'Camb. Phil. Proc.,' vol. 2, p. 286.

t 'B.A. Reports, Cambridge,' 1904, p. 472.

PRODUCED BY HOT PLATINUM IN DIFFERENT GASES. 61

The only general conclusion which these numbers point to is that the work required
to liberate a positive ion tends to be smaller than that required to liberate a negative
ion. This is probably due to the fact that the negative ions are corpuscles, whereas
the positive ions are associated with a certain amount of matter.

Throughout this paper, the view has been maintained with regard to the negative
ioiiisation that it is due to the escape of corpuscles which are present in a more
or less free condition insido the metal, the variations in the amount of negative
ioiiisation caused by different gases being due to the effect the gases have on the
amount of work necessary for a corpuscle to get through the surface. The positive
ionisation, on the other hand, has been supposed to be caused entirely by traces
of gas absorbed by the metal. There is, however, another view of the origin of
the positive ionisation which cannot be lightly dismissed. This is that it is due
to the escape of positive ions which are moving freely inside the metal in much
the same way as the negative corpuscles have been supposed to be, and that the
effects of different gases are simply due to the changes they produce in the
surface. There are a number of considerations which make this hypothesis plausible
at first sight. It would give an obvious explanation of why the positive obeyed
the same temperature law as the negative ionisation, and would also account for
the work necessary for a positive ion to esca{>e being in general smaller than that
for a negative ion. For since the particles at the surface of the metal are all at
the same temperature, the energy of a moving ion will tend to become equalised
with the average value for the metal at each collision, so that, to take an
exaggerated case, an ion which had found its velocity reduced to nothing, owing
to the work it had done in getting through a certain fraction of the surface, would
get a fresh start if it made a collision. Thus the work function in question will
l>e equal to the average amount of work to be done between the last collision
inside the metal and a point outside ; it will thus be greater for an ion with a long
free path, such as a corpuscle, than for one with a short free path. On this view, the
effects of different gases are to be explained by the changes they produce in the work
required for an ion to escape from the surface layer. This might occur by the
formation of an electrical double layer, or simply by a change in the physical
properties of the bounding region, or by both. The former would act differentially
on the positive and negative ionisations, whilst the latter might be expected to act in
the same way, though not necessarily to the same extent, on both. A combination of
the two could obviously be made to account for any observed simultaneous change in
tlir two leaks. If we calculate the number of ions per cubic centimetre inside the
metal from tin- ionisation in different gases according to the formula previously* given
by the author, we get the numbers given in the columns N+ and N_ in the last table.
These mnntars do not mean much, owing to the ignorance of other data which really
enter into the calculation, but on the view at present discussed they should range

* 'Camb. Phil. Proc.,' voL 2, p. 286 it Kq.

62 MR. 0. W. RICHAEDSON ON THE IONISATION

around the true value for the number of free ions per cubic centimetre. This would
give for the positive ions 1018 and for the negative 10S1 ; thus the number of free
positive ions would be insignificant compared with the number of free negative ions,
and the theory would at once account for this small value for the amount of one
metal transported into another by the electric current.*

To test this theory, which has both simplicity and elegance to recommend it, both
the positive and negative ionisations were measured simultaneously in oxygen at low
pressures. The very great increase in the positive ionisation, produced by small
quantities of oxygen as compared with other gases, would indicate, if we accept the
above view, that the effect was probably specific and caused by the formation of an
electrical double layer. In this case, increasing the positive should decrease the
negative leak. The results of this experiment, which are given on p. 22, show that
although the oxygen altered the positive leak in a ratio of 10 to 1, the negative leak
was not changed by 20 per cent. As it is very unlikely that some other effect of the
oxygen would compensate so well as this, the writer considers that this experiment
renders the above view very improbable. It is chiefly on account of this experiment
that the view, that the positive ionisation is due to the absorbed gas, and only
indirectly to the metal, has been adopted in describing the results obtained in this
investigation.

X. — § 17. SUMMARY OF THE PRINCIPAL RESULTS.

The positive ionisatiou, i.e., the number of positive ions produced by 1 sq. centim.
of platinum surface per second, possesses a minimum value, which depends on
temperature and pressure, in most gases. The positive ionisation in oxygen at a low
pressure (< 1 millim.) is much greater than in the other gases tried. In oxygen at
low pressures, and temperatures below 1000° C., the ionisation varies as the square
root of the pressure ; at higher temperatures and low pressures it varies nearly
directly as the pressure, whilst at higher pressures at all temperatures the variation
with pressure is slower, so that at pressures approaching atmospheric the ionisation
becomes practically independent of the pressure.

The variation with pressure in air is similar to that in oxygen. In nitrogen and
hydrogen the ionisation appeared to increase more rapidly with the pressure at high
pressures than in oxygen. In very pure helium at low pressures there was a positive
ionisation which was a function of the pressure.

The experiments on ionisation by collisions indicate that the positive ions liberated
by hot platinum in oxygen are of the same order of magnitude as those produced by
the collisions. They are not great masses approximating to dust particles.

The positive leak in oxygen always oscillated round a certain value under specified

This view can easily he made to give a reasonable quantitative explanation of the change in the
positive ionisation produced by oxygen at different pressures.

PRODUCED BY HOT PLATINUM IN DIFFERKNT GASI-s 63

conditions. It was, therefore, never steady, so the minimum values were taken.
This kind of effect was much less marked, if it occurred at all, in the other gases.

The minimum value of the positive ionisation was found to remain practically constant
with a wire heated during three months at various times, for 1 50 hours altogether, in
oxygen at 900° C. to 1000° C. Moreover, four different wires of different dimensions,
after continued heating in oxygen, gave nearly the same value for the ionisations
at the same temperatures and pressures.

The positive ionisation in air at constant temperature is smaller than that which
would be obtained if the nitrogen were withdrawn, so as to leave only oxygen at a
low pressure. The nitrogen, therefore, exerts an inhibiting effect on the oxygen.

The minimum value of the positive ionisation at a definite pressure in all gases
appears to be connected with the temperature by the relation first deduced by the
author for the negative ionisation. This relation may be written t = A0*e~<J/a*, where
t is the ionisation, 6 is the absolute temperature, and A and Q are constants. The
value of the constant Q, which is a measure of the energy associated with the
liberation of an ion, is in most cases smaller for the positive than for the negative
ionisation.

These results refer to wires which have been heated in a vacuum and sul>sequently
in the gas in question for a long time. New wires exhibit peculiar properties,
especially in regard to their behaviour under different electromotive forces. Old
wires also exhibit hysteretic effects with change of pressure and temperature.

The view is developed that the positive ionisation is caused by the gas adsorbed by
the metal and the consequence examined of supposing the ionisation to be
proportional to the amount of the adsorbed gas present. In the case of oxygen, by
making the assumption that the rate of increase of the adsorbed gas is proportional
jointly to the concentration of the external dissociated oxygen and to the area of
unoccupied platinum surface, whilst the rate of breaking up is proportional to the
amount present, a formula is obtained which agrees with the experimental results.

This formula is that the ionisation t = -. * , where p = (kP+±k3)*— %k, P being the

external pressure and k the dissociation constant of oxygen. A, B and k are
constants depending on the temperature and are of the general form a0*e~*'. Thus
this view accounts for both the temperature and pressure variation.

The positive iouisation from the outer surface of a hot platinum tube in air is
increased when hydrogen is allowed to diffuse through from inside the apparatus.
The increase in the ionisation is proportional at constant temperature to the quantity
of hydrogen escaping from the surface in unit time. For different temperatures the
effect produced by a given quantity of hydrogen is greater the higher the
temperature.

The negative ionisation from hot platinum in air is unaltered when hydrogen is
allowed to diffuse out through the platinum.

L <* J

II. Second Memoir on thr Compositions of \ttmbtrs.
By Major P. A. MAcMAHOx, R. A., D.Sc., F.R.&

Received August 23,— Read December 6, 1906.

PREAMBLE.

bi a Memoir on the Theory of the Compositions of Numbers, read before the Royal
Society, November 24, 1892, and published in the 'Philosophical Transactions' for
1893, I discussed the compositions of multipartite numbers by a graphical method.
The generating function produced by the method was of the form

a symmetrical function of the quantities a.

The investigation of the present paper leads, in part, to the same generating function
which is subjected to a close examination. Moreover, the whole research has to do with
the compositions of numbers, and appropriately follows the Memoir of 1 893.

The problem under investigation, which was brought to my notice by Professor
SIMON NEWOOMB, may be stated as follows :—

A pack of cards of any specification is taken — say that there are jt cards marked 1.
<l cards 2, r cards 3, and so on — and, being shuffled, is dealt out on a table ; so long at
the cards that appear have numbers that are in descending order of magnitude, they
are placed in one pack together — equality of number counting as descending order-
but directly the descending order is broken a fresh pack is commenced, and so on until
all the cards have been dealt. The result of the deal will be m packs containing, in
order, a, b, c, ... cards respectively, where, n being the number of cards in the whole

(ate...)

composition of the number ?«, the numbers of parts in the composition being m.
We have, then, for discussion—

(1) The number of ways of arranging the cards so as to yield a given composition

(ate...);

(2) The number of arrangements which lead to a distribution into exactly m packs.
These problems, and many others of a like nature, are solved in this paper.

VOL. OCV1L— A 414. K 21.1.07

66 MAJOR P. A. MA. MAHON ON THE COMPOSITIONS OF NUMBKKS.

The first of the two questions has given rise to two new symmetric {'unctions,

of great interest, which supply the complete solution. The second gives rise to the
same generating function .that presented itself in the first Memoir. It is here
attacked by the calculus of symmetric function differential operators, and a number
of new results obtained.

If the whole pack be specified by the partition

there is a one-to-one correspondence between the arrangements which lead to a
distribution into m packs and the principal compositions, involving m— 1 essential
nodes, of the multipartite number

~~

Part I. is concerned with an elementary theory of the case in which the cards are
all numbered differently.

The general case, which is more difficult, is dealt with in Part II.

To make what follows clear to the reader, I commence with some elementary
notions concerning the connection between the partitions and compositions of
numbers on the one hand, and permutations and combinations of things on the other
hand, and I also specify and describe the nomenclature and notation that I have
found it convenient to adopt. A suitable notation is, indeed, of the first importance
in this subject, as I hope to make evident as the investigation proceeds.

INTRODUCTORY.

Art. 1. Any succession of numbers, written down from left to right at random,

such as

142771,

is termed a " composition " of the number which is the sum of the numbers.
If the numbers be arranged in descending order from left to right,

774211,

the succession is termed a "descending partition," or simply a "partition" of the
number which is the sum of the numbers.

Or, if we arrange in ascending order of magnitude,

112477,

the succession may be termed an " ascending partition."

Generally, in speaking of partitions, we understand that the descending order is
meant ; but it is convenient sometimes to consider them as being defined by an
ascending order.

MAJOR P. A M \.M.\IION ON THE COMPOSITIONS OF NUMBERS. 1,7

There is no other method of ordering a collection of numbers which is of ^ru
application.

We see that the same collection of numbers gives rise to only one partition, but, by
permutation, to more than one composition.

Art. 2. Both partitions and compositions have an appropriate graphical represen-
tation. That of a partition was first given by FERKKI:S. and the notion was elaborated
by SYLVESTKK during the time he was at the Johns Hopkins University in Baltimore,
U.S.A. It consisted merely iti writing a row of nodes, or units, corresponding to each
number (or part) of the partition, the left-hand nodes of the rows being placed in a

vertical line. Thus

774211

is denoted by

Art. 3. A trial will show that this method is not suited to compositions. One
method, effective for certain purposes, was given by the author.* To indicate it,

consider the composition

142

of the number 7.

— * — . — . — . — * --- .

We take seven segments on a line, and place nodes, *, so that the line is divided
off into 1, 4 and 2 segments respectively in order. The conjugate composition is
reached from this by suppressing the existing nodes and placing nodes at the points
of division which are free from nodes.

Thus

. — . — * — * — * — . — * — .

denotes the composition 21121

•

Art. 4. There is a more illuminating mode of representation which is here given, it
is believed, for the first time ; it is akin to the method of FERRERS, and enables
methods of research which SYLVESTER'S exertions have made familiar.

It consists in taking rows of nodes in order and placing the left-hand node of any
row vertically beneath the right-hand node of the previous row.

Thus

142
is denoted by

* " Memoir on the Theory of the Compositions of Numters," ' Phil. Trans. Roy. Soo.,' 1893.

ic 2

«i« MAJOR P. A. MAOMAHON ON THE COMPOSITIONS OF

aud ' 142771

by

This graph is read horizontally ; the conjugate is obtained by reading vertically,

giving

21122111112111112,

or, in brief notation,

2122al82152.

We may also read the graph horizontally from bottom to top and vertically from
right to left, obtaining generally four compositions from the graph.

The graph is a zig-zag one and will be, without doubt, an important instrument of
research.

PART I. — SECTION 1.

v

Art. 5. Consider the permutations of the first n integers, and for simplicity
take n = 9.

Writing down a permutation at random,

31|4|5|92|76|8,

it is clear that lines can be drawn separating the numbers into compartments in such
wise that in each compartment the numbers are in descending order of magnitude.
We can then write down a succession of numbers which describe the size of the
compartments, proceeding from left to right, and thus arrive at a composition

211221
of the numl>er 9.

I say that the permutation under examination has a descending specification

(211221) or (212221).
Similarly, from the ascending character

^

3 | 1459 | 27 | 68

of the same permutation, I say that the ascending specification is

(1422) or (1422),

where it is to be noticed that 1422 is the composition of 9 which is conjugate to
2la22l, the composition which specifies the descending character. This is shown by
the zig-zag graph

MAJOR P. A. MA.MMIOX « >N THK COMPOSITIONS OF NUMHKl;- »;«»

Art. 6. We can now formulate the question : Ot the permutations of the first n
numbers, how many have a descending specification denoted hy a given composition
of the number n ? Whatever the answer, it is clear that the same answer must, in
general, be given for three other compositions, viz., the three others associated with
the zig-zag graph. In fact, from

314592768 of specification 211221,
we derive 867295413 „ 2241;

and from these two by changing the number m into n—m+ 1,

796518342 of specification 1422,

243815679 122112

and so forth.

In two cases there are two associated compositions instead of four, viz. :—

(i) When the composition reads the same as its inverse (that is the same from

left to right as from right to left),
(ii) When the conjugate and the inverse are identical, as in 221, whose

conjugate is 122.
*The numlier of self-inverse compositions of an even numl>er 2m and of an uneven

number 2w+l is

& .

The numl>er of inverse-conjugate compositions of an uneven numlier 2w-t 1 is

2".

Hence, in the present theory, the number of different numbers that appear in the
case of an even number 2m is, since the whole number of coiujxisitions is 2*""',

£.2" + i(2*— '-2-),
= 2"-*(2"-I+ 1);
and, in the case of an uneven number 2»i+l,

viz., it is 2»-*+21/8("~4>

according as n is even or uneven.

* See " Memoir on the Theory of the Compositions of Nurnlxjrs," ' Phil. Trans. Roy. Soc.,' 1893,

70 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

Art. 7. Let N(«&c...) denote the number of permutations of the first n integers
which have a descending specification denoted by the composition

(ale...)
of the number n.

Obviously N («)=!, a = n.

To determine N (ab), a + b = n, separate the n integers into two groups, a left-hand
group of n numbers chosen at random and a right-hand group of the remaining b
numbers. This can be done in

( ) different ways.
W

I write — —. r-7 = 1 j in a common notation ; now arrange each group of

numbers in descending order of magnitude for each of the ( ) separations ; we thus

obtain each of the permutations enumerated by N (a, b) and the one permutation
enumerated by N (a + b).

Hence

or

~ \aj \a + bj \a

Again, to find N (abc), we separate the n integers into three groups containing
a, b, and c integers respectively ; this can be done in

a\b\c\

different ways ; placing the numbers in each group in descending order., we obtain all
the permutations enumerated by

N (abc), N (a + b, c), N (a, b + c), N (a + b + c).
Hence

N («6c) + N (a + b, r) + N (a, 6 + c) + N (a + b + c) = ^[^ ;
leading to

nl nl

T

a! 6! c\ (a + b)l c! a!
where a+b + c = n.
Similarly we find

n! n\ n\ n\

N (abod) =

a\b\c\d\

(ct+b)l(c+d)l
where a + b + c + d = n,

n ! n! n ! n !

M \.loi; !'. A. MvrMAHON ON THK COMPOSITIONS OF NUMBERS. 71

The general law is clear; the letters ", l>, <•, d are always in order in the
denominators and the sign of a fraction depends upon the number of factors in its
denominator.

We can thus calculate the number of permutations appertaining to each of the
2""1 compositions of n.

It has been established independently, by the aid of the zig-zag graph, that these
numbers M- / \

are equal in four's or in two's.

Art. 8. The sum of the numbers N (...) is of course n!
The details of the above results for

n = 2, 3, 4, 5, 6
are given for easy reference.

N(2) = 1 1

N(l') = 1 1

2 = 2!

N(3) =N(13) = 1 2

N(21) = N(12) = 2 4

6 = 3!

N (4) = N (I4) = 1 2

N(31) = N(13) = N(21') = N(1'2) = 3 12

N(22) = N(121) =5 10

24 = 4!

N(5) =N(1&) = 1 2

N(41) = N(14) = N(213) =N(132) = 4 16

N(121») = N(1!I21)= 9 36

N(318) =N(la3) = 6 12

N(2*l) =N(12*) = 16 32

N(131) =N(212) = 11 22

120 = 5!

7-j MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

N(6) = N(i") = i 2

N(51) =N(15) = N(214) = N(142) =5 20

N(42) =N(24) = N(1821) =N(1213) = 14 56

N(32) = N(12212) = 19 38

N(41a) =N(124) =N(313) = N(133) =10 40

N(141) = N(2122) = 19 38

N(321) = N(123) = N(2212) = N(1222) =35 140

N (312) = N (213) = N (1231) = N (131*) = 26 104

N(132) = N(231) = N(2121) = N(1212) = 40 160

N(2S) =N(1221) =61 122

720 = 6!

Art. 9. Some simple summations are obtainable from elementary considerations.
In regard to the permutations of the first n integers, let

' ''"'.•*•• 2N (*...),

where «<n, denote the sum of all numbers N (...), such that s is the first number in
the specifying composition. Take any s+1 of the numbers

1, 2, 3,...w,

and arrange them from left to right in such wise that the first *• numbers are in
descending order and the s+ 1th number greater than the sth ; this can be done in

n \

+ 1J ways;

the remaining n— s— 1 numbers can be arranged in (n— s— 1)! ways, so that, placing
them to the right of the former, we arrive at the result

SN («...)=- n!
Art. 10. Again, denoting by

the sum of all numbers N (...) of which the specifying compositions commence with
exactly *— 1 units, the consideration of the properties of conjugate zig-zag graphs
establishes that SN (!"'...) = SN (•...),

with a single exception where s = n ; e.g.,

[...)= SN(2...) = 2^,
and so on.

No restriction is placed upon the number next to the unit in this case,
t Here the number following the unit must be > 1.

MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 73

Art. 11. Again, for the summation

where the composition begins with fit leant x units, we easily obtain the value

n\

The Multiplication Theorem.

Art. 12. A fundamental property of the numbers N(...) will be established from
elementary considerations ; it will, later on in the paper, be generalised.

Let

N(o1o>...ok)

be derived from the permutations of p different integers, and

N («.««.+»•••«.•«)
from the permutations of n— p different integers ; it is to be shown that

where on the right the reference is to the permutations of n different integers.
Out of the n numbers

1, 2, 3,...n,
we can select

(
numbers in

ways, and arrange each selection, so as to have a descending specification

(«!«,... a.),
in

N(rr,a,...a.) ways;

the remaining numbers can be arranged, to have a descending specification

(«, + l«.+J. ••«!+«)»

in

N («.+1a,+1...a,+,) ways ;

placing the latter to the right of the former there appears

arrangements.
VOL. ccvu. — A.

N Ka»-«.) N («.

74 MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS.

Now, combining the two sets of numbers, we find that either there is or there is not
a break in the descending order between

a, and «,+, ;
hence the number of arrangements is also

.rts+() + N(«1a2...a.-i, a,+as+l, aI+2...«,+(). Q.E.D.

Art. 13. Regarded as a numerical theorem, the multiplication is commutative, but
in regard to form it is not commutative ; thus, by considering the multiplication

(a,+1a,+,...e^+«) N («!«.,...«,),
we obtain the linear relation

Observe also that the order of the numbers in brackets in any number N(...) can
be reversed at pleasure and thus new forms of results obtained.
As a verification : from the tables

N (12) N (11) = N (1212) H- N (131) = N (132) + N (122) ;
10 . 2 . 1 9+11 4+16

N (123) + N (15) = N (312) + N (42).
35 + 5 = 26+14

The fact that the multiplication is not commutative formally is of great importance
in the theory of these numbers.

Art. 14. Extending the theorem to the product of three numbers

N («!«». . .a.), N (&A. ..bt), N (cjC.,. . .cu),
we find

'

+ N(aj. ..«.&!• ••&<-!, ^t + ^l, C2...CU) + N(«i-. .«,-!, «.+ />!, />2--.^-l, ^ + Ci, C2...CU).

We may, in general, give the right-hand side 3 ! different forms corresponding to
the 31 permutations of the numbers N(...) on the sinister.

If we take the product of m numbers N (...), to form the dexter, we combine the last
integer of a number N (...) with the first integer of the next following number N (...),

MAJOR P. A. MAcMAHOX ON THE COMPOSITIONS OF NUMBERS. 75

0 times iii 1 way ,

/T»-l\

f l j ways,

hence 2*"1 numbers N (...) present themselves on the dexter.

Not counting reversals of order, the dexter can, in general, be given as many
different forms as there are permutations of the numbers N(...) on the sinister.
Counting reversals, the number of different forms is further multiplied by 2", subject
to a diminution when one or more of the numbers N (...) is self-inverse.

Applications of the Theorem.

Art. 15. The theorems, already arrived at above, are particular cases of multipli-
cation. Thus the formuhe, of which

N (abc) + N (a + 6, <•) + N (a, b + c) + N (a + 6 + r) = -^Lj
is a type, are equivalent to results, of which

£j^Tj N («) N (6) N (c) = N (abc) + N (0+6, r) + N (a, b + c) + N (« + b+c)
is representative, since j^ /fl\ = j^ m _ N (c) = 1.

That the sum of all numbers N (...), of given weight n, is >* ! is shown by the formula

since on the dexter occurs an N(...) corresponding to every composition of n.

Art. 16. Suppose that it is required to find the sum of all numbers N (...), of given
weight, which are such that each associated composition commences with a given

series of numbers

alat...am,

or, in other words, suppose we wish to make the summation indicated by

the solution is given at once by

for, by the multiplication process, the unit which terminates N (nla,...am\), combined
with "-*1-1

76 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

gives every composition of the number

n-2c*.
Hence, since N (1) = 1,

Art. 17. By varying the order of the factors, on the sinister of the multiplication
formula, a variety of interesting results present themselves ; thus

Ul

where after am, on the dexter, occurs every composition of

n—ta—p;

and the portion

...a

includes every composition of

which terminates with a number not less than a^
Hence, for such a summation,

a formula which is independent of p.
Art. 18. In particular from

{N(l)}-2a-1N(a1«2...«ml)
we obtain

SN (...a\a2...aml) = ^ JS(a,a,...aml) ;

wherein the summation is for every composition of

n— aa— ...— am— 1

which terminates with a number not less than c^.
E.g., for n = G, «, = 1, a, = 1,

N (412) + N (1312)+N (2212) + N (18213) = |JN (211).

10 + 26 + 35 + 19 = 6.5.3
Art. 19. As another example of the power of the theorem, let

(the numbers a,, o8...aOTj, bly ha...bmi being given) denote a summation in regard to
compositions of _^ _^,

MAJOR P. A. MM MAHOX ON THE COMPOSITIONS OF NUMBERS. 77

placed between amt and &, ; we obtain
SN (ata,. ..«„,.. .&!&,.. .&.,)

n!

n!

By varying the order of the factors, other summations, leading to the same
numerical result, can be effected.

Art. 20. Consider next the multiplication

(x, + 2)!(*,+2)!(x3+2)!
x {N (I)}"--1 N ( l"+a) {N (1)}«*-*N (1"+J) {N (I)}1"'-' N ( r<+») {N (I)}"--' ;

wherein, Si^ + S* = «,

Wi, wa, w3) wt are numliers not less than unity,
*i. sa, "a are any numbers, zero not excluded.

The result of the multiplication consists of numbers N (...), such that there is

(i) A composition of u't followed by x, units, succeeded by
(ii) A composition of wa followed by xa units, succeeded by

(iii) A composition of w:t followed by .s3 units, succeeded by

(iv) A composition of w4 ;

and the dexter is the sum of all such nuinliers N (...).
Denoting this sum by

SN (w^W WH04),

we find that its value is

n!

since each number N(...) occurring in the product on the sinister has unity for its
value.

Hence, in general, the remarkable theorem,

showing that the sum depends merely upon the numbers

78 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS,

and not at all upon the numbers

Observe that wl and the final number of the composition may or may not be unity,
and that every composition of n may be written in the form

If

3=S=S'= =0

mtwwww ) = --nl

2!'

and, in particular,

wherein iv2, ;r3, ... wm_, are non-unitary, but wit wm may or may not be unitary.
As a simple example take

u\ = 1, s, = 4, w2 = 1,
so that

a verification.

Art. 21. A more general theorem is yielded by

n!

wherein

are given integers and the summation indicated on the dexter is in respect of the
whole of the compositions of the numbers

WL «'a, «>3, w4,
where

03=?<?, — 1. wa— 2, w3—2 and u\ — l.

The value of the sum is thus

MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.. 79

which, by the multiplication theorem, may be given the form

+ N ( ty, . • ./>., 1 '</, . . .

+ N ( \p, . . .pm>2qi . . .c/^r, . . .rm\ )].

Evidently, from the above, comprehensive results can be obtained from the
multiplication theorem.

SECTION 2.

Art. 22. The next problem I propose to solve is that of determining the number of
the permutations of the first n integers, whose descending specifications contain a
given number of integers, or, in other words, whose associated compositions involve a
given number of parts. The solution is implicitly contained in a paper I wrote in the
year 1888.*

Let N., denote the number of permutations associated with compositions containing
exactly m parts.

In the paper quoted, I had under view a collection of objects of any species — say
p of one sort, q of a second sort, r of a third, and so on — and defined the objects as
to species by these numbers placed in brackets. I thus formed a partition

(pqr...)

of the number n, such partition being the species definition of the objects.

As equalities may occur between the numbers p, q, r, ..., I took, as a more general
definition, the partition

(pfW~),

where zirp = n.

In the case under consideration, where the integers (or objects) are all different,

the species definition is the partition

(I').

I proved, in the general case, that the number of ways of distributing the objects,
into m different parcels, is given by the series

" /m+pt-l\* /m+p3-l\"
\ P, ) \ P* I '

(m\ /m+p1-2\'1 /m+pt-2\*' fm+pt-2\'»

"\ ft

(m\ /m+^,-3\- /m+^,-3\" /m+p,-3\"
Wl Pi ) \ P, ) \ P* I

" Symmetric Functions and the Theory of Distributions," • Proc. L. M. S.,' Tol. »*., p. 22«5.

80 MAJOR P. A. MAcMAIION ON THE COMPOSITIONS OF NUMBERS.

For the case in hand, p^ = 1,77] = n,

Art. 23. I shall prove that

For consider the arrangements enumerated by Fm. Place the compartments (or
parcels) in order, from left to right, in any one such arrangement, and, in each
compartment, place the integers in descending order of magnitude. The arrangement
is obviously one of those enumerated by

Nm) Nm_j, Nm_2, ... or N,.

In the whole of the arrangements, enumerated by Fm, thus treated, each arrange-
ment enumerated by Nm will occur once only.

1 | 2 | 3 | 4 | \rn-l or m—s.

. ._. . .......... -_.

Let the illustration denote an arrangement enumerated by Nm_!. Each segment
denotes an integer, and the m— 2 vertical lines separate the integers into com-
partments.

By placing an extra vertical line at one of the unoccupied points of division, we
obtain an arrangement enumerated by Fm. This can be done in (n— 1)— (m— 2)
different ways, showing that the particular arrangement, enumerated by Nra_j, is
derivable by obliteration of a vertical line from

n— m+l

different arrangements enumerated by Fm.

Hence, the forms Fm include the forms NOT_j each n— m+l times.

Again, let the illustration denote an arrangement enumerated by Nm_,. By placing
s extra vertical lines, at unoccupied points of division, we obtain an arrangement
enumerated by Fm. This can be done in

i'n—m+s\

\ * /

different ways; showing that the particular arrangement, enumerated by Nm_,, is
derivable, by obliteration of ,s- vertical lines, from

n—m+s\
s

different arrangements enumerated by Fm.

MAJOR P. A. M.\. MAHOX ON THE COMPOSITIONS OF NUMBERS.
Hence the forms F,,, include the forms Nm_. each

(n— m + 8\ ,.

1 times.

Hence

FM = NM+(H-™+I)NW_I+(W-:

Thence it is easy to show that

and also

The relation SNW = n! may he verified.

Art. 24. It follows at once, from the zig-zag graphs, that

N. = N._M+I.
Some of the simplest results are

81

n

N,.

N,. N,.

N4. N..

Na.

1

1

2

1

1

3

1

4 1

4

1

11 11

1

5

1

26 66

26 1

6

1

57 302

302 57

1

Art. 25. There is another interesting series for N,,.
Let

denote the expansion ot

when deprived of the term which is linear in p and of the term independent of p ;

and put

P. = (l+*)"-,.o ;
then

N -P -

-1

1 V
'

1 prove a general theorem, of which this is a particular case, later on in the paper.
VOL. ccvii. — A. M

82 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

Art. 26. Considering next p different numbers, defined by the partition

we have, by a previous definition,

where a1} a2, o3)... are each <1 and such that

S« =p.
I have written

NP>

instead of NM, in order to specify the number of objects (or numbers) subjected to
permutation.

Art. 27. I shall now prove that

where in

the number of objects subjected to permutation is n, and the summation is in respect
of all permutations such that the sum of the first m numbers in the descending
specification is equal to p.
For, by Art. 16,

2N («!«.,...«„,...) = "' N (a^... awl) ;
hence

and, by the multiplication theorem,

(jp+l)N(a1cra...am)N(l) = N(alaf...ft»l) + N(a,a,.. .
so that

i •
and since

SN

«

whence, by summation,

•

but since

MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. 88

so that, substituting,

hence

Art. 28. Further, summing each side with respect to m,
SSSN(«,aa...e/n...)

,n n"

n!

but the sinister is of the form

SN (w'.Wj) (see Art, 20)
and thus has the value £« ! ; hence

an interesting result.

Art. 29. From a previous result

hence

2N(a,rtj...am+l) = 2N

n n

and it may be observed that the miml>er8, included in

are the conjugates of those included in
Art. 30. Also since

,aa...oM+l) = Nm,,i'-n-SN(a1cf,...om_,l),

n a

2N (aIai...a.-,

a

and this leads to the relation

E.g., for p = 3, m = 2, p-ni+2 = 3,

3N3,1,+2N,,1, = N,.,.
verified by

8.1 + 2.4 = 11.

M 2

84 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

The result is convenient for the calculation of the numbers Nmjlp+i from the
numbers Nmilp.

We have also the remarkable result that the probability of obtaining a permutation,
such that the sum of the first m numbers of the descending specification is p, is
independent of n, and has the value

(jH-l)I m'l'
whenever p is n — I or less.

Art. 31. From the definition we have in respect of the permutations of n numbers

N1+N2+N3+... = n!
I shall now show that

2
nl 6\

the summation being for all values of

"i, "a, •••

such that

zv = 0,

'Ssv, = n.

The theorem is the outcome of the multiplication theorem of Art. 12.
Observing that, for all values of s,

N(l«) = 1,
we have

!'•) N ( 1") N

*l I f<2 '. #3 !

+ N
and generally, for the product

since sv, = n,

= a linear function of numbers N ••••

(i !)" (2 ;-'... (

We may write down a similar result for every permutation of the factors of

and, by addition, obtain
n!

= linear function of numbers N,

where Si> = 6.

MAJOR P. A Mv MAHON ON THE COMPOSITIONS OF NUMBERS. 85

Further, we obtain a result of this nature for all values of

such that Sw, = n,2v = 0; and, by addition, we obtain

ST .?! — ;— K- — r— ^ r = linear function of numlien N,

where Sw. = n, 2»> = 6.

We have now to determine the linear function of numbers N which appears on the
dexter.

If one such number be j^ / ^ \

it is evident that / » \

is some composition of the number n.
Consider the product of 6 factors

N(1")N(1*)...N (I1*),
where S* = n.

The process of multiplication produces N numbers of 6 different kinds.
In the first place we throw all the units together,

N (l"+VK" + '*)(

one N number containing n parts.

In the second place we combine a consecutive pair of factors and throw the
remainder of the units together, thus producing 6-1 N numbers each containing

n — 1 parts, viz.,

N(l"-121

N i*'+*«~1

In the third place we combine two consecutive pairs (including, of course, a
consecutive three) of factors and throw the remainder of the units together, thus
producing ,e_

( 2

N numbers each containing ?i-2 parts, viz., the series of which one is

N (l''-'
Notice that, if *, = 1, this becomes

8fi MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

We proceed in this manner until finally we combine 0—1 consecutive pairs and
throw the remainder of the units together, thus producing

N numbers, each containing n— 0+1 parts.

Hence the compositions that present themselves are included in those enumerated by

N,, N»_,, , NB_9+1.
We have to consider the product

in all of its permutations and for every system of values of

•V> 'S2> •••> «S9>

such that

•Sj + .Sjj +...+.<(«, = n.

Hence, from considerations of symmetry, and attending to the modits operandi of
the multiplication theorem, we find that the whole of the compositions enumerated by

N N N

•"•I ^n-l) •••> -^n-S-M

present themselves.

Hence the linear function we seek is a linear function of

N>J "NT "NT

n-fl + l) L~n-6+2> •••> "»— II -L"*)

and it remains to determine the coefficients.

The number of products, including permutations,

N(1'-)N (!")... N (IV),

•

which we have to consider, is ecpual to the numbers of compositions of n into 6 parts,
viz., it is

(n-e) >

each of these produces

S0-V

m

N numbers, each containing n— m parts.
There are thus

\n-6l \ m

N numbers, each containing n—m parts.

But there are only

{n— 1

m

MAJOR P. A. M.v. MAHON ON THE COMPOSITIONS OF NUMBERS. 87

different N numbers, each containing n—m parts, because

m

is equal to the number of compositions of n into n—m parts.
Hence, each N number, comprised in

N.-., '
will occur

w;^n-m -r)tirae8-

it — v

("»')

Hence the required linear function is

v/n~ m~ 1\ M

or

and the final result is

X n!

where

Sw, = n, Si> = ^.

PART II. — SECTION 3.

Art. 32. In the preceding pages we have had under view the permutations ot n
different numbers. As I am now taking in hand the general case of numbers which
possess any number of similarities, I find it convenient to slightly alter the point
of view.

Let «, Ay, -.

denote numbers in descending order of magnitude, and suppose there are

p number equal to a,

so that, placed in descending order, the assemblages may be written

I say that the assemblage is sj>ecined by the composition

(pqr...).

88 MAJOR P. A. MAoMAHON ON THE COMPOSITIONS OF NUMBERS.

As equalities may occur between the numbers p, q, r, ..., I take, for greater
generality, the specifying composition

(PW-)-

It will be seen later that the order of occurrence of the parts of this composition is
immaterial, so that we may consider the parts plt pa, ... to be in descending order of
magnitude and the specification to be denoted by a partition

tew-)-

E.g., we obtain the same results for each of the six assemblages,

aa^yyy, a/3/3/3yy, ctfiftyyy,
the specification of each assemblage being

(321).
Every permutation has a descending specification.

has the descending specification

In the case considered in Part I. the assemblage of numbers had the specification

(1-)

since there were no similarities, and the numbers N(...) were expressed in terms of
the coefficients obtained by the multinomial expansion

(a1 + o3+a3+...)n.
E.g., we found

N (a) = coefficient of symmetric function (a) in the expansion,

where, in the first case, « = n, and in the second, a + b = n.

In a usual notation let

«i, fh, fh, •••

denote the homogeneous product sums, of the successive orders, of the roots of the
equation ---' --'-+ .... = 0 ;

we may say that, in Part L, the auxiliary generating function was

(a1 + a2 + a3+...)'1 = V,

au «s, aa, ••• being the roots of the equation.

Art. 33. In the present case the auxiliary generating function is

f) *>}> *>/) *»
... np, "P, "A ••••

as will appear.

P. A. M^MAHON ON THK coMpasiTioNs OF NUMBERS. 89

For it was shown, lor. cit., that the number of ways of distributing the objects,
as specified, into different parcels containing a, b, c... objects respectively is the

coefficient of the symmetric function

(abe...)

in the development of the symmetric function

as a sum of monomial symmetric functions.
Let this coefficient lie denoted by

and let the number of arrangements of the objects, which have a descending
specification

be denoted by

Let the whole number of objects be

tirp = n.
Then, when a — n, clearly

N(a) = C(a)= 1,

and when a + b = n, C(ab) is the number of arrangements into two different parcels
containing a, b objects respectively, and by previous reasoning

and, when a + b+c = n,

N (dbc) + N (a + 6, o) + N (a, b + c) + N (a + b + c) = G (abc),

and so forth as in the simple case already considered.
Hence

N (oft) «C(ofr)-O (<>+&),

N (abc) = C (abc)-C (a+b, o)-C (a, b+c)+C (a+b+c),
N (abed) = C (abcd)-G (a + b, c, d)-C (a, b + c,d)-C (a, l>, c+d)

Ac.,

the numbers N being all expressible iu terms of coefficients of the auxiliary generating
function.

Art. 34. E.g. Take objects aaa/8/Jy, where a, ft, y are in descending order of
magnitude.

Since

hMi = (6) + 3 (5l) + 5 (42)4-8 (413) + 6 (3*)+ 12 (321)

+ 1 9 (3 18) + 1 5 (2») + 24 (21!*) + 38 (2 1*) + 60 ( 1«),

VOL. CCVJI. - A. N

90 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS,

we calculate, from the above formulae,

N(6)=l, ' . ' -

N (51) = 3-1 = 2,
N(32) = 6-l = 5,

N(321) = 12-5-2-1 = 4
and so on.

The five arrangements, enumerated by N (3a), are

OLa.

.ya.pfi

a/3/3aay
/3/3yaaa,

each having the descending specification (3a).

The four arrangements, enumerated by N (321), are

each having the descending specification (321).

The complete results qud numbers specified by (321) are

N (313), N (133), N (214), N (142), N (2112) }

>each 0 . . 0
N (1213), N (1321), N (12212), N (I6) J

N (6), N (412), N (124), N (1312), N (1231)|
N (2212), N (1222), N (2121), N (1212) J

N(51), N(15), N(312), N(213) „ 2 . . 8

N (141), N (1221) „ 3 . . 6

N(42), N(24), N(321), N(123) „ 4 . , 16

N(32), N (231), N (132) „ 5 . . 15

N(2:1) „ 6 . . 6

60

60 being, of course, the total number of permutations of the objects.

MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBEBB. 91

Art. 35. The method of calculation establishes that the number N (...) is unaltered
by reversal of the order of the numl>ers in the bracket.

Also that the results are only dependent upon the magnitudes of thr parts in the
specification of the assemblage and not ujxm the order of their occurrence.

General Investigation of a Generating Function.
Art. 36. I have shown above that, for numbers specified by

(K'K1-),

an auxiliary generating function is

W-.

for, from its expansion in terms of monomial symmetric functions, the numbers

can be succeasively calculated.

For present convenience I take the above generating function to be

and recall that N(a6p...) + N(a+6> f> ...) + N(a, 6+c, ...)+...
is equal to the coefficient of symmetric function

(abc...)
in the expansion of

The above linear function of the numbers

N(...) .
is formed by adding adjacent numbers

0, 1, 2, 3, ..., k at a time,

where the numbers a, b, c, ... are k in number.
It thus comprises 2*"1 terms in general
Art. 37. Let this linear function be denoted by

so that if we write = w(abe...).(abc...)t

From this system of linear relations is determined the set
N (a) = C (a), where a = n,
N (ah] = C(nb)-C(a+b), where a + b = n,
N (abc) = C(o&r)— C(a + 6, c)— C(a, 6 + c) + C(a + 6+c), where a + 6+c = n, and soon;

N

92 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

the law of formation of the linear functions of the numbers

C(...)
being similar to that which occurs in

M(a)(6)(c)...},

with the exception that the signs are alternately positive and negative, depending
upon the numbers of integers in the brackets.

Art. 38. Denote this linear function of the numbers C(...) by

fc{
8othat N (ate...) =

When it is necessary to put in evidence the numbers whose permutations are
under examination we may write the two formulae

ey{(a)(b)(c)...}{pqr.., = C(abc...)(pqr..,;

SECTION 4.
Digression on the Forms 0y , </>c .

Art. 39. Define in general, so that

0N {(<*!. . .a.^a.) (&!&.,.. . &«_ A) (ciC2. . .cm-iCm}^dt. . .de). . .(A^. . .k,)},

where there are k symbols a, b, c, d, ..., k, denotes the 2*"1 terms forming the series

N (<*!...*,)

.&,_i, &«+Ci, ca...kz)
+ ...

+ N (a,. ..a,-i, a.+bt, b2...bt-l,
+ ...,
where additions take place,

0, 1, 2, ..., k— I at a time between the pairs a,, b^ ; bt, Cj ; CB, c/, ;
Art. 40. Similarly define

<£c {(<*i.. .«,_!<*,) (&i63. . .6»-A)(c1Ci...C^1c.)(d1df. . .dv). . .(k^. ..k,)}
to denote the 2*"1 terms forming the series

!...«,_!, a.+bi, &,...*,)
C (a,.. .&,_!, 6, + Cx, c,...k.)

a1...a,_1, «'. + &,, ba...bt-i,

MAJOR P. A. MACMAHON ON THE CO^OSITIONS OF NUMBERS. 93

formed according to the same law, but the successive blocks of terms having
alternately positive and negative signs.

Art. 41. I proceed to generalise the two results

By definition

6V {(«,... a._,a.) (6,6,... 6,)}

j.-.&^ + N (a,..
and since

this

Now the sum of these two terms is precisely

because the terms involving ,

are the same, with opposite sign, as those involved in

and therefore cancel them.
Hence the result

Art. 42. Again

t_1, a.

by successive use of the formula Art. 4 1 above.

94 MAJOR P. A. MAcMAHOSrON THE COMPOSITIONS OF NUMBERS.

Also, clearly, if t = I

= <£c {(a,). ..(a.-i)(«A<i
Art. 43. Therefore, by induction, we can express any form

<M }

as a form

<M }•

The law is well seen by a particular case, viz.,

ex{(a)(b)(c)(d)}=<t>c(abcd),
0*{(ab)(c)(d)}=<i>c{(a)(bcd)},

eti{(a)(b)(cd)}=<t>c{(abc)(d)},
0N{(a)(bcd)}=<j>c{(ab)(c)(d)},

occur

0*{(abcd)}=<f,c{(a)(b)(c)(d)}.

We have, in respect of the four letters, 8 = 23 relations ; the letters always

in the order

a, b, c, d,

and to obtain the form <f>c{ }, which is equated to a form #N{ }, we may make use
of the zig-zag conjugate law; e.g., connect with

(ab) (cd)
the composition 22 ; take the zig-zag conjugate of this, viz., 121, and then write

0x{(ab)(cd)}=<j>c{(a)(bc)(d)},
and

M(«)(k) (<*)}=*>{(«*)(«*)};
and so in every case.

Art. 44. In the general case of p letters we obtain 2f~l relations corresponding to
the 2*""1 compositions of p ; the relations are obtainable from zig-zag conjugation of
such compositions and, in any relation

we may interchange the form- symbols

#}»> $C-

Art. 45. In the above investigation we obtained incidentally certain linear relations
between the forms

MAJOR i'. A. MA.-MAHON ON THE COMPOSITIONS OF NUMBERS. 95

and also between the forms

9ci

which must now be set forth in a regular manner.
The former relations are of the type

this follows directly from the definition of the form 0N{ }, since

0K(a&c...) = N(
Art. 46. The latter relations are of the type

which also follows directly from the definition of the form <f>c{ }, since

4»c(et&c...) = C(afcc...).
Art. 47. We have other linear relations of the type

In fact, the law may be taken to operate as between any sets of consecutive

tor8lQ , ^N{ } and

leaving the remaining factors untouched.

factor8lQ , ^N{ } and ^c{ } respectively,

96 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMHKHS.

Thus it is easy to verify the three relations

0»{(ab)(cd)(ef)(gh)}
= 0y{(abcd)(ef)(gh)}

and the further three

8»{(ab)(cd)(efgh)}
+ 0x{(ab)(cd)(e,f+g,h)},

e*{(abcd)(efgh)}
+ es{(abcd)(e,f+g,h)}
+ 0s{(a, b + c, d)(efgh)}
+ 0N{(«, b + c, d)(e,f+g, h)} ;

<t>c{(abcd)(ef)(gh}}
-<f>c{(a,b + c,d)(ef)(gh)},

4>c{(aV)(cd)(efgh}}
-<f)C{(ab)(cd}(e,f+g, h)},

4>c{(abcd)(efgh)}

-<j>c{(abcd)(e,f+g, h)}

+ <^c{(a, b + c, d)(e,f+g, h)}.

Art. 48. From these relations we may obtain new relations by transforming from

to <£c, or vice versd.

Thus from relations of type

we obtain those of type
<£c(a&c) = <j>c{(
and from those of type

} = <j>c(abc)-<j>c(n + b, c)-<j>c(a,
we obtain others of type

These new expressions for

6y(abc...) and <f

with an obviously analogous law to that we have frequently met with, are of great
importance.

MAJOR P. A. M\. \i.\irON ON THE COMPOSITIONS OF NUMBERS. 97

From the relation

we obtain

9»(abcd) = M(a)(&c)(rf)}-M(« + &, c)(d)}-0,{(a)(b, c+rf)}+^{(a + 6, c+d)} ;
and there is no necessity to give further examples.

SECTION 5.

Art. 49. The differential operator, of order s, that is so frequently of use in the theory
of symmetric functions, viz. : —

can now be employed.

Remembering that operating upon monomial symmetric functions,

Da(a)=l,

Da (f>) = 0 unless b = a,

DaDJ)e...(a&c...) = 1;

and generally that Da obliterates a number a from the partition of a function and
causes it to vanish if no such number presents itself, it is clear that

a A t» V '* D'D>D<

and thence if we write

according to a law derivable from that which defines

(«) (&)(e)...} (see Art. 38),
we find

Art. 50. Observe that in the paper to which reference has been made it was shown
that

Two consequences flow from this fact.
Firstl

which is a theorem of reciprocity for the numbers

N (...).

VOL. CCVII. —A. O

98 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

Secondly, since

...)(Mr...) = DpD?Dr...

where, on the dexter, the operand is a function formed from the functions /i,, h3, h,3, .

in the same manner as

#0{(«)

is formed from the operators

Hence
where

Art. 51. I now write
so that

and it appears that

Ninitr \ _
I CfrUC . . . J (pqr.. .) —

is the true generating function of the numbers

for the permutations of assemblages of numbers of all specifications.

In fact,

h^... = 2N (a&c...)(MP...> . (pqr...} ;

and the expansion of

hate...

as a linear function of monomial symmetric functions gives a complete account of
numbers

Art. 52. Before proceeding to a rapid examination of this new and most important
symmetric function ,

" abc • • • 1

never before I believe introduced into algebraic analysis, I give complete tables of the
numbers N (...) as far as n = G.

n = 2.

(2)

(I2)

N(2)

1

1

N (1*)

1

= specification.

MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

99

n = 3.

(3)

(21)

(!')

N(3)

1

1

1

N(21)

1

2

N(l>)

1

specification.

n = 4.

'(*)

(31)

(22)

(21s)

(I4)

N(4)

1

1

1

•

1

N(31)

1

1

2

3

N(2*)

1

2

3

5

N(121)

1

2

5

N (21s)

1

3

N(l<)

1

- specification.

n = 5.

(5)

(41)

(32)

(31-)

(2*1)

(21s)

(1s)

N(5)

1

1

1

1

1

1

1

N(41)

1

1

2

2

3

4

N(32)

1

2

3

4

6

9

N(131)

1

2

3

6

11

N (2s!)

1

2

4

8

16

N (31s)

1

1

3

6

N (212)

1

2

5

11

N(121S)

1

3

9

N (21»)

1

4

N(l')

1

specification.

ioo MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

n = 6.

To explain — it will be found that

corresponding to row 4 of the table for n = 5.
Art. 53. Another symmetric function

is formed from the elements

in the same manner as the symmetric function

from the elements

(6)

(51)

(42)

(32)

(412)

(321)

(23)

(31s)

(2*P)

(21')

(I6)

N(6)

1

1

1

1

1

1

1

1

1

1

1

N(51)

1

1

1

2

2

2

3

3

4

5

N(42)

1

2

a

3

4

5

6

7

10

14

N(32)

1

2

3

0

5

6

7

9

13

19

N(141)

1

1

2

3

4

6

7

12

19

N(231)

1

2

2

5

7

9

13

23

40

N(312)

1

2

3

5

7

14

26

N(321)

1

1

2

^

6

8

11

20

35

N(23)

1

2

2

6

10

11

18

33

61

N (412)

1

1

1

3

3

6

10

N (3P)

1

1

4

10

N (1221)

1

3

6

6

13

28

61

N (2212)

1

2

3

6

15

35

N(1312)

1

2

3

5

12

26

N(2121)

1

3

3

7

17

40

N (2122)

1

2

7

19

N(12212)

1

2

C

19

N(1213)

1

4

14

N(21«)

1

5

N(l«)

1

{specifica-
tion.

MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS. 101

SECTION 6.
The Symmetric Functions Aftftft , (tflflfl... .

Art. 54. These two new functions are of fundamental importance, not only in this
investigation, but in the theory of symmetric functions generally.
In regard to the algebraic equation

AI, A,, l>n, ... are the homogeneous product sums of the roots and the two sets of
elements

have reciprocal properties which it is useful to briefly glance at.

We have

/,, = «, = (!),

A3 = al3-2a1a3+aa = (3) + (21) + (I3),
and, in general,

The two series of elements are connected in such wise that, in any relation between
the elements, the symbols a, k may be interchanged. Thus, from

a,a-3aa= -2/h3

is derived

A, — 3A3 = — 2«1-

As a particular case it is found that a, is the same function of the elements
hi, AS, /J3, ... that h, is of the elements a,, aa, «3, —
If functions of the elements /iu A,, fi3, ... be denoted by

we see that, if

then

showing that

is an absolute invariant qud the transformation which replaces the elements

AI, A,, A,, ...

by the elements

J

102 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OP NUMBERS.

Art. 55. With these necessary preliminary remarks I define a new function of
weight 71, viz. : — ,

where pipapa--- is any composition of the number n ; of the given weight there are

such functions, one of which is clearly

The complete definition is given by the multiplication law

= h +/i

where the functions , ,

are, or are not, of the same weight.
Art. 56. A second new function

is similarly defined by the same law ; viz.,

What follows applies generally to both of the new functions.

Art. 57. Since the multiplication is commutative, we have the first important
property, viz.,

PiPi — P-9t9i—fi '"Pi —Pi- \,P-+9i,

~ "' ... i- - >"'

p-

Art. 58. Every product of elementary functions is expressible in terms of the new
functions, e.a.,

and in general

where, in 0h{ }, the sum of the coefficients is

2-1.

These relations show that , ,

fipip*~p- ~ "p.-.

Art. 51). Similarly

-

and in general

It, moreover, we define

2...qt) (r}r2...ru)}

MAJOR P. A. M \.M.\HON ON THE COMPOSITIONS OF NUMBERS. 103

as denoting

i

,...Mi- 1*t...r. + "•*...»-,, P.+1,, f,...fir,...r.

and

as denoting

according to the law usual in this subject ; we find

"ft ...p,hq, ...jAv- r. •"">,... p.. ,,*+7i. tt-lfl'i '•
— ''-- "'••• -

<
= fa { (Pi

= <M (pi.

= ™h—Mk

The reader may verify that

*»{(P) (9) W}.
have each the same value W,_

which we have denoted by ,

"rv

Art. 60. I pass on to drag into the light some important relations connecting

hPl...r, and «,,...„.
When the relation

/, - V/—V+*1 (^0' «*. «*•

ft, - i(-) ,-. -yta» •••"«
AI :. ..«,.

was under ol>servation just now, it will not have escaped notice that this is precisely
the expansion of

«n = «ia-«2,
am = a,s— 2a,a3 + a3; &c.,

and by the law of formation we see that

a,. = /<„ ,

and theuce 7

a, = A,..

104 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

The known value of hn is thus given by a law identical with the multiplication law
of this paper, and the expression of AB in terms of

is completely given by

,

fln — C£j».

This new statement, of a well-known law, immediately suggests the generalization
to which I proceed.

Observe that n and 1"

are zig-zag conjugate compositions.
From the relation

is now deduced

and, since , ;

and we again observe that

pq and \p I

are zig-zag conjugate compositions.

Hence writing (1^21^) = (pq)',

a(py> = "(PI)' '•
and, in general, I have established (but reserve the proof for another occasion) that

where / \ , v

(Pip-i---), (PiP*-.-)

are zig-zag conjugate compositions.

Art. 61. The theorem has an interest of its own, but it is also of vital importance
in this investigation. This importance consists partly in the circumstance that the

functions

'W...

are those which naturally arise in the present theory of permutations. The present
theorem enables the immediate expression of them in terms of the elementary

symmetric functions

»i, «2, a3)...

and thus they may be more easily dealt with by symmetric functions differential
operators. In fact, the homogeneous product sums

can be made to disappear from the investigation ; but, as will be seen, it is sometimes
advantageous to retain them wholly or in part.

MAJOR P. A. MAOMAHON ON THE COMPOSITIONS OF NUMBERS.

ios

Art. 62. To gain familiarity with the new functions I give without proof some of
their elementary properties.

where *, is the sum of the nih power of the roots.
The following expression for afl»-«

a.i"- = fi.-ih*-.+i-<*.-Jin-,+
The result of operations with Df, viz.,

If .<?„,« denote the sum of the symmetric functions whose partitions contain exactly
t parts, we have the companion tables, in which the law is obvious.

Vi-

«kt-

tn.».

V4-

Vs.

«„.«•

a,-

1

+ 1

+ 1

+ 1

+ 1

+ 1

On-'

1

+ 2

+ 3

+ 4 +5

ttsi">

1

+ 3

+ 6

+ 10

«41"4

1

+ 4

+ 10

«»!"•

1

+ 6

<*er-'

1

a,-.

Oj,-«.

a,,-'.

a41~.

0H-*.

««"•.

Vi

1

-1

+ 1

-1

+ 1

1

«».»

1

- 2

+ 3

-4

+ 5

v»

1

-3

+ 6

-10

«".4

1

-4

+ 10

*».s

1

- 5

««,«

1

The fundamental properties of these new symmetric functions were communicated
by me to Section A of the British Association for the Advancement of Science, at the
York meeting, 1906, August 1-8.

Art. 63. The generating function of N (nbc...) is either

hoi*... or a(ate y.

VOL. CCVII. — A. P

106 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

We can now determine the highest symmetric function, in dictionary order of the
parts, which occurs in the development of halx . This, by the known theory of
symmetric functions, is obtained from the form

a(ate...)'

by expressing (abc...)' as a partition and taking the Ferrers conjugate (abc...)" ;
then we see that no symmetric function, prior in dictionary order to

(abc...)",
can appear.

Also the highest integer in (abc...)'

is the lower limit of the number of parts, occurring in the partition of a symmetric
function, arising from the development of

E.g., since

,

"141 ~

we arrange 2122 as a partition, obtaining 2212, and taking the Ferrers conjugate from
the graph

we reach (42) as the highest symmetric function in dictionary order that occurs in
the development of hin.

Hence N(l4l)e = N(14l)M = 0.

(See the table of weight 6. )
Numerous relations such as

^i4i + ^5i = hiv+ha

can be verified by the same table.

Art. 64. Before proceeding to establish the multiplication theorem, the generali-
zation of that in Part I., it is necessary to examine the mode of operation of the

differential operator p.

*/•

upon a product , ». Wl

npi np, •••>

vv»»

It is clear that T. 7 ;

1>A = hp-a .

In the paper it was shown that

where a' a" denotes a composition of a into two parts, zero not excluded, and the
summation is for every such composition.

MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 107

Hence

E.g., D4M»=<

where the compositions of 4 have been taken in the order

40, 31, 13, 22.

In general T) / / / ?/> h I

where / « <«>

is a composition of a into a or fewer parts.

It is to be noted that in forming the compositions zeros are parts, so that, for

iastance' 400, 040, 004

count as different compositions.
If the operand be

since i\ / \ i

*Ja(ctfl) = 0 unless a — 1,

we need only attend to the compositions composed of units and zeros.
Thus

It is easy to show that
from which

and, particularly,

N(4^)1 =

7 4+2+1

from the table.

Similar formulae can be established at pleasure.

The Conjugate Law.

Art. 65. It has been seen (Art. G) that, when the numbers permuted are specified by

I",

where

(pq...),

denote conjugate compositions.
We write the theorem

N(^---)a-)

and we may inquire into the existence of an analogous theorem when the numbers
permuted have any other specification.

P 2

108 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

Consider the expression ; 7

n(pg...) — 'hf9...Y>

which is the generating function for the difference between

N(pq...) and ~N (pq...)',

for all specifications of the numbers permuted.
The generating function may be written

^P7...~°W..>

according to the theorem proved above.
The differential operation ^

has the equivalent forms

hence n „,

A "»-

is the same function of , , ,

flj, fla, ha, ...

that

D!^...

is of

«1, «2, «3, ••••

It follows at once that ^. „ ,, \

Di"(*»--«W-) -

equivalent to the known result

already found.

Art. 66. Now, considering the generating functions

or

must be of the form » , T>L a

(A + 2B){(2) + 2(12)}.
Hence r» »-sr

...
equivalent to

N (p9...)(21...0+N (pq.-.Y^^ = N Qjgr.. .)(,.

Thus, from the table n = 6,

(3
13 -f 6 = 19.

MAJOR P. A. MAC-MAHON ON THE COMPOSITIONS OF NUMBERS. 109

Art. 67. Again, operating with

m

we obtain a result -of the form

A (/<,-o,)+ B (Mi -o/»i)

(A + B){(3) + (21)};

hence ^.-^ (/,„_«„) = D.-D. (*„

equivalent to

and, particularly, from the tahle

N (321)(W-N (2*1 V) = N (321)<«,-N (2'!%,,,
8 3 = 20 15.

No new result is obtained by taking

as the operand.

Art. 68. Further, Di"*(V. -

has the form
reducing to

equivalent to the new result

and, particularly, from the table

18 13 11 - 6.

Art. 69. If we take here the operand to be

a new result is obtained, viz.,

The above is sufficient to indicate the nature of the results which present them-
selves ; I have not attempted to generalise them. The question appears to be a
difficult one.

110 MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS.

SECTION 7.

Generalisation of the Multiplication Theorem.
Art. 70. I will establish the result

where the summation is for all solutions of the diophantine equations

\ + ri+... = la,

=P,

For consider

N°W

the summation being for all solutions of the diophantine equations

r,+r2 = r,

Moreover,

and, by like reasoning, the theorem as enunciated follows.

MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. Ill

As examples, N(321)(-)+N(33)<-) = 3N(32)an),

derived from <?N { (32) ( 1 ) } (aw ;

and N(24)(aM)+N(231)(aj)+N(213)(M2)+N(222)(af)

= 6N(21)(m)+18N(21)(M),
derived from

Art. 71. The enumeration of the permutations, whose specifications contain a given
largest integer, will now be investigated.

Let T .1 K

AM) "mi •rv-»

denote respectively

in which

(i.) the highest of the integers a, 6, c, ... w m or less ;

(ii.) „ „ ,, .- or greater;

(iii.) „ „ » exactly;

so that, when a + b + c+... = n,

Iw =

Jm.= KM
I. = J, =

Im — Im-l = "m~»m + l = •*•!

0H(n) = N (n) = J. = K. = I.-T.-, ;

. = J.-.+J. = -I.-.-I.-.+al. ;

'118O

)»} = N(n-2, l') + N(n-l, l) + N(n-2, 2) + N(n),

, n-l) + N(n),

l, n-
and by addition 3^w_2)(l)8} = K._,+2K._,

the law apparent here obtains so long as a number n-v appearing in

N( ),

112 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

on the right-hand side, is not equal to any other number in the same bracket ; so
that, when *<%n, (g+ J} ^ {(n_g) (} }>}

Hence

Jm = («
Km = (n-m+1) 0N{(m) (l)— }-2 (n-m)

and, the specification of the numbers permuted being

(pqr...),

0N {(a) (b) (c) . . . } = C (abc. . .) = DJ)6DC.
thence T

or, m not being less than the greatest integer in £ (it + 1 ),

is the generating function of the number J,,, .
Similarly ^

and, m not being less than the greatest integer in \ («+ 1),
(n-m+ 1) /<m/t1"-m-2 (n-wi) ft*+A""*~1t(»-W

is the generating function of the number Km.

Similarly, but subject now to the condition that m must not be less than the
greatest integer in £ (n— 1),

(„_,„_ |) /-,n+A"~""2-(«-m) A.+A"'"'1
is the function which generates the number

Subject to the conditions mentioned, we have a complete solution of the problem,
but when m has other values, the solution is less simple and I see no way of
effecting it.

MA. TOR P. A. ifACMAHON ON THE COMPOSITIONS OF NUMBERS.

113

SECTION 8.

Art. 72. I recall that the number of ways of distributing numbers (or objects)
specified by (nWtv''Dw> )

into m different parcels, is given by the series

\Pi)\frJ\P»

/m\ (m +pl — 2\'' fm +pt— 2\'' (m + pa— 2\*1

~uA PI ) ( P, •) \ p*

. fm\ fm + />, - 3 V" fm + »,- 3\'- fm + »8- 3 V1
h\2/\ *./('*)•(*) '"

and this, for brevity, I write

v

*n,

Let

1
\ * /

denote the number of distributions, associated with a descending specification
containing exactly m parts, and write this

N.,

when there is no risk of misunderstanding.

Following the proof of Art. 23, it may be proved that

/n-rn+r

and nlso

N = F -I

A1 m — •*• m I

and thence

XT =G— ^tt"^MG +(^"MG +( — }m*l/ """" A \G

Art. 73. From this relation the following results are obtained : —

n = 3.

(»>

(21).

(I*)-

N,

1

1

1

N

2

4

N.

1

VOL. COVII. — A.

114

MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

n = 4.

4

1

(31).

(22)-

1 * • <

N, 1

1

1

1 1

N, :

3

4

7 11

N3

1

4 11

N4

1

n = 5.

(5).

(41).

(32).

(31').

(2*1). (2P).

(P).

N,

1

1

1

1

1 : 1

1

N,

4

6

10

12 18

26

Ns

3

9

15 33

66

N4

2 8

26

N5

1

n = 6.

(6).

(51).

(42).

(41*).

(32)-

(321).

(31').

(2»).

(22P).

(214)-

(I6)-

NX

1

1

1

1

1

1

1

1

1

1

1

N2

5

8

13

9

17

25

20

29

41

57

Ns

6

16

9

33

67

48

93

171

302

N4

1

9

27

20

53

131

302

N5

1

*

16

57

N6

1

To explain, observe that the number at the intersection of the row N3 and the
column (2812) shows that ^ . , = 93.

These tables will be of constant service in verifying results to be obtained.
Art. 74. From the relation

•* = ^m— i JOT-1 o

\ L 1 \ * /

we can obtain a system, for, summing each side from m

Nm+Nm_1+...+N, = G.-

= 1 to m = m,

MAJOR P. A. MA.MAHOX ON THE COMPOSITIONS OF NUMBERS. 115

•*

and, repeating the summation 6 times,

so that, when 6 = n,

Again, taking differences instead of summing, we get the series

XT \r c /n+2\r /n+2\r

JNm-JNM_, = ^»~( ! )<-s,-i+( 2 J«L_t

N«-2N.

and in general

These restilts are all given by the two formulae

which become the same when p = 0.

Curious Expression for
Art. 75. I shall now prove that

N _p m-2/7t\ m-3/u\

N»- p-»-«z

where

P.- ^-

o

denotes the expansion of

Q 2

116 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

when deprived of the terms linear in plt p3, ..., and of the term independent

For it is easy to show that two consecutive terms

N, in— t— I in\ T> ./ \t+im— t— 2 / n

, ym-t-n-p ,

~^rW ""'-1

may be given the form

, \tm— t— 1 fn\ /pi+m—t—l\"' ip2

~

+m—t—I\''

,(

Pl p,

, w« + m-<-2/n \ fpi+m— t-3\'< (p3+m— 1-3\'*
H m_l -(t + l)( Pl p2

and, giving t the values 0, 2, 4, ..., and summing and simplifying, we obtain

ipi

! + m — 1 \" ipi + m - 1
\

Pi \ P*

(n+l\

2

v

which we know to be the value of Nm.

Art. 76. The symmetry of the numbers Nm _/ will not escape the notice of the
reader.

SECTION 9.

Art. 77. My purpose now is to connect the preceding pages with my Memoir on
the Compositions of Numbers, to which attention has already been directed. In the
course of that investigation I had occasion to consider the permutations of the
letters in a?p>y,

with the object of determining the number of permutations containing given
numbers of £a contacts,

If we take any permutation

,../Ja...ya...y/8...y/3a...

and particularly notice all of such contacts, it is clear that the numbers of parts
in the descending specification a, j3, y, ..., being numbers in descending order of

MAJOR P. A. Mv.-MAIlnX OX THE COMPOSITIONS OF NUMBERS. 117

magnitude, is necessarily one greater than the numtar of such contacts ; in the
present instance there are 6 parts in the descending specification and 5 contacts. The
problem of the determination of the pmiiutiitions having descending specifications
containing m parts is identical with that which is concerned with those lmving»m-l
contacts of the nature specified.

Art. 78. I established in the Memoir that the letters in

can be permuted in

\ »n J \«*
ways so as to have exactly ^ ^ contacts>

% 7*

••>•» y£
and I further discovered that this number is the coefficient of

in the development of the function

(a + A^jS + Any)' (a + £ + A^y)' (a + ft 4 y )'.

Art. 79. In the same paper I showed that for this function may be substituted
the function

which does not involve p, q, r, and may therefore lie regarded as the general
generating function of the numbers.

Art. 80. Reserving for the present the generalizations, which were also given in
the papers referred to, it is clear that the application to the present question is
obtained by putting

AJI = Aai = Asa — A>

when we find that the number of permutations of

«>£V,
which have descending specifications containing m parts, is the coefficient of

in the development of (

or of 1

I - (a + £ + y) + ( 1 - A) (a/3+~ay +£y)-(l - A^y

This, therefore, is the true generating function of the numbers N».

118 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

It may be verified, for example, that the complete coefficient of

•

(1 + 20X+48X2+20X3+X4),

which agrees with a previous result.

From a previous result also the coefficient of

fm + p— 1\ fm+q—l\ /m + r— 1\
( f )( I )( r )

/m+p-2\ /m + q-2\ im+r-2\
\ 1 A P )( <! )( r )

+ 1\ /m+p-3\ im+q-3\ /m-f>-3\
2 A P )( 9 )( r )

where n = p + q + r.

Art. 81. Observe that the generating function is a symmetric function of a, yS, y,
verifying a previous conclusion that an Nm number is not altered by any interchange
of the letters a, ft, y.

When the numbers p, q, r are equal, that is when the objects are specified by the
partition ,,.

we can establish a symmetrical property of the numbers N.

For coefficient

X"-l(a.pyy in (
is, by writing

— for X and Xa, Xy6, Xy for a, ft, y,
X

equal to coefficient of

X3p-ra+l(«£y)p in (X

equal to coefficient of

^-"^(oLfty)" in

equal to coefficient of

in (a + X/3 + Xy)* (a

Art. 82. Hence M _ M

Nm — JNjjp-m+S»

and the numbers N range from >T xr

INi to lN2p+i,

showing that 2p+l is the maximum number of parts in the descending specification,
when the objects are specified by the partition

MAJOR P. A. MA.MAHON ON THE COMPOSITIONS OF NUMBERS. 119

Art. 83. In general, when there are /• different letters,

the number of permutations of

which have descending specifications containing m parts, is the coefficient of

in the development of

or of

This is the general generating function of the numbers

N..

Art. 84. Since it is symmetrical in regard to

a,, a,, ..., at,

the value of Nw is not affected by permutation of the letters

«i, «s, ... , at.

Art. 85. It can be shown also, as in the simpler case, that when

/>i = P*= ••• = /'* = />.

the coefficient of \»-

A

is equal to the coefficient of

--

so that -»T _ -»j

S*m — i>(*-i

the numbers N range from M ,.,

IS, tO JN^-i

and (k—\)p+l is the maximum number of parts in a descending specification.

SECTION 10.
Art. 86. The generating function

now presents itself for examination.
Introducing the elementary functions

a,, a,,

120 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBEES.

and writing 1 —A = b, it is written

1— c^ + fcoa-frV, + ...+(-)* b*-la* '
I

1-A'
where

A = a1_
For the present purpose we may consider k to be infinite, and write

A = «! — bn2+ 62a3— ... .
Art. 87. Taking the symmetric function operators

, = ai1aj203 ... = ,

and an auxiliary fictitious equation

r being an infinite number, it is necessary to remind the reader of the relations
existing between the operators.

Successive linear operations of , 7 ,

dto a,, «,, ...

are denoted by placing them in separate brackets, thus,

but when they are multiplied, as in TAYLOR'S theorem, so as to produce a single
operator of higher order, they will be placed in one bracket, thus,

Art. 88. Let monomial symmetric functions of the fictitious relation

af-D^-' + DjZ'-'-... = 0
be denoted by a partition in brackets with subscript D, thus,

( )D-
Then I have shown, in a previous paper,

and, in general,

d, = U,2-2D2 = (2)D,
d, = . . . = (s)D,

MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS. 121

Art. 89. Every symmetric function identity has corresponding to it a relation
between the operators ; thus corresponding to the set

(I)4 = (4) + 4(3l) + 6(2*) + I2(21a) + 24(l*),
we have the set

= 2 (l')D+(2)D,
,)« = (</,«) + 3 (rfAJ+rf, = 6 (l«)D + 3 (21)D+(3)D,

= (24) (!«)„+ 12 (21i)D + 6 (2% + 4 (31)D + (4)D,
and so on.

Art. 90. Also, corresponding to the set
2a, = »,*-•«»,
6as = Sj*— 3*,.'».l

Ac., we have the set

2D, = (</,') = (<W-d,t

24D4 = (d,4) = (d,)4-<
and so on.

Art. 91. For the special operand

1

1-A

these operator relations assume a special simple form which is of great importance in
the theory of the generating function.

For , . / \._ii._i /, j A \ / \.-ii.i-i

rf,A = ( — )' 16' (1— bA.) = (—)' lb'

or, qud the above operand, , _ (_y-it)'-ifi

and thence, from a set of relations given above,

Art. 92. By means of these we can now arrive at a most important series rf
relations.

VOL. ocvn. — A. B

122 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

(pq)D = (dpdg) =

^ = (dfdgdr)

and generally
and more generally
or, if STTJ? = ?i, STT = t,

Art. 93. From the relations

s!D
we find the set

D1a = 2D,-6D,,

D,8=6D3-66D2+&aD1,

D,4 = 24D4-366Da+14//D,-6>D1,

Art. 94. And also the set

MAJOR P. A. Mw-MAHoX o\ THE COMPOSITIONS OF NUMBERS. 123

The. ExpressilnUty of D,.
Art. 95. The fundamental relation

exhibits D, in terras of powers of DI.

It is clear, A pinori, that D, is expressible in terms of D, and powers of D,, e.g.,

) (3) D» = D' (D'Dl + 8W)»+ 96'Dl + 1 2fti)»
(2) (4) Da = Da(D^+^I),DI-f-29^D,+24i3D1 + 3064), and so on,

where notice, as a verification, that the siim of the numerical coefficients is the same
on the two sides.

In every case Ds appears as a factor.

In general the operator products, which appear on the right, are factors of

WDk,

which contain the factor D3, every weight of ojrarator product being represented once,
ciiid once only, from the weight 2 up to the weight of the single operator on the left-
hand side.

It is important to remark that /o*i\

\* /
is a perfect partition* of the number

2/fc+l ;

because every lower number can be composed in exactly one way by the parts of the
partition.

Art. 96. It will now appear that there exists an expression for

D.

corresponding to every perfect partition that can be constructed.
The general expression of a perfect partition is

where a, /8, y, 8, ... are any positive integers, zero excluded.
The perfect partition ,~^ \

is the particular case

« = 1 , p = A\ y = 8 = . . . = 0.

' Messenger,' 1890, p. 103.

Eo
m

124 MAJOE P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

In every case, if <r be the highest figure in the perfect partition, D,, is a factor of
the expression for D,, e.g., taking the perfect partition

3*l2,
40D6 = 2D32+3&DSD12+15&8D3D1 + 20?/DS.

I do not interrupt the investigation by stopping to prove the theory of expressi-
bility depending upon perfect partitions ; its truth is intuitive.

Art. 97. It is necessary to labour the subject of the operator relations, qud the
special operands, because the whole theory of the numbers Nra is involved.

Art. 9£. Perhaps the most interesting of the operator relations are those which do
not involve b (or X).

Recalling the relation of Art. 92, viz.,

where _, _,

ztrp = n. ZTT = t,

we may also write

J n

where

and, if

n-i = v-j,

we may eliminate b, obtaining

Art. 99. The simplest formula thence obtained is found by putting

and this leads at once to , , 7 ,

rijrfs-rt/ = 0,

D2D12-4D/+3D3D1 = 0,
which also results by elimination of b from

Art. 100. To obtain spme more relations in a simple manner, I write

MAJOR P. A. M\< \I.\HON ON THE COMPOSITIONS OF NUMBERS. 125

•

and then (*+«)D(«»)D = («)D(«+")D = (OD(« + «)D I

or, as these relations may be written,

d.+t du = d, dt+* = dt du+, ;
(s+t, tt)D = (*, *+«)„ = («, u+*)D,

with the usual multiplier (viz., 2), if either

x 4- / = •'*, or f + « = .«, or « + « = £.

Art. 101. We are led to the series

(81)0-2(2%,

(41)D = (32)D)
(51)D = (42)D = 2(3')D,
(61)D = (52)D = (43)D,
(71)D = (fi2)D = (53)D = 2 (4a)D) Ac.,
and generally if (^/'...), (^,«-...)

be functions of the same weight and degree, viz.,

Application of the Foregoing to the Genwatiiig Function.
Art. 102. It has been established that

We will first of all examine the result of the equivalence of operators

2!D, = D1'+(1-X)D,
(see Art. 93 qud the operand on the right-hand side). Write the operand

m = 3,

2N3,, = N8.

verified (from the tables) by

2.48 = 93 + (l5

126 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

•

Art. 103. Again, in the same formula, put

r, = n-2, c2 = 0, r3 = r., = ... = 0,

Wefind 2N..a.-« = N^ + fN.-N,.-,)."-!.

We obtain, from this, a useful result by writing

n— m + \ for m,
for then 2N.-.+I,«-2 = NB_m+1.1» + (NB_m+1-N,_mV.-i.

Art. 104. Observe that XT -,.T

Nm,i» = JNn_m+lil»,

so that by addition and subtraction we obtain

m+li21'.-2 = Nm-1»

or, as we may conveniently write these relations,

(N.+N.-.+On— = N«.,« :
(Nm-NB_m+1)2i"-2 = (N7n-NB_B1+1)i'-1 :
= (N.-N.-O,-!.

These are the relations connecting Nm and N,_m+i qud the subscript 21"~a analogous
to those connecting the same symbols qud the subscript 1".

Art. 105. From any operator relation we can immediately derive a relation between

the numbers Nm by substituting for

b'VV...
the expressions

and this it is convenient to denote by

N(<r)m, ,-,/....
Art. 106. Thus, corresponding to the operator relation

6D3 = Vl(Dl
we obtain

As a particular case put

c, = n-3, rz = r3 = c4 = ... = 0,
so that

MAJOR P. A. MAC.MAHON ON THE COMPOSITIONS OF NUMBERS. 127

and thence

(Nm-N._-+I)«-s = (N.-N.-O."'-

For n = 6, these relations can be verified by the tables for all values of m,
Art. 107. Similarly the theorems derived from

can be at once written down.

| It is worth noting that this operator relation can, by putting D, = /;A,, l>e written

W...D.D.,...

we can write down the corresponding relation between the numbers N,
It will be found that /xr

is a linear function of vr vrp)

**m. l"i •1* n«,l »

Hnd (N.-N...*

a linear function of N'Vi-', N^ ,.-»,

and that the same obtains when instead of

•V

we take ,.-1.

•vl-xj • • • 1 •

Art. 108. From the operator relation

where tt,,, «i, «-,, . . . are numerical coefficients that may be determined.

Thence is derived the relation u „, ,,

^".ih r* •••

giving a hint to put N »„.,-« = N.-'1 symbolically,

and then /xr •> \w, /M i \»i

- - ' 8ymbolically.

Art, 109. It must now be remarked that, since

128 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS.

we obtain

and since XT XT XT,. t / UXT

Nf.,-« = N,,_,_e+lil'-<, N;;i,+I = (-)'N,,

and

/NmWNroy'_.
we obtain ^,

_ /Nn_,,t+1WNn_M+ly"
' ( P> ) ( P, )

Art. 110. We have two alternative expressions for

in terms of numbers

I verify them in the case

J

+ 3(Nm-Nm_1)15

agreeing with, for m = 3,

8.48 = 302 + 3(66-26) + 3(ll-2.11 + l) + (l-3. 4 + 3.1).

NMi!,=

and, for m = 3,

= N4>1. .

-3(N4-N3)1S
+ 3(N4-2N,+N,)1.

agreeing with

8.48 = 302-3(26-66) + 3(l-2.11 + ll)-(-3.1 + 3.4-l).

MA Jon P. A. MAC-MAHGN ON THE COMPOSITIONS OF NTMI:I i> 129

Art. 111. We have seen that, in general, we have two expressions for

but, since ^ _ jj

we have four expressions for -^

* M P*

viz" /Nw+«-

P
~ . (Nr/,

( '*P+P )*

(V)'

Art. 112. It is clear that the operator relations afford unlimited scope for obtaining

theorems connecting the numbers

N... „-,,,«

Relations, so far utilised, have involved the operator

D,,

but it is easy to construct them so as not to contain D, and generally so as not to
contain D. , where $ is less than a given integer.

E.g., from the symmetric function relation

we find

D,' = &SD,-6&D3+6D4;
and generally the relation

leads to

or, throwing out the factor

D,' = fc1),-6&D3+6D4,
the same relation as before.
Moreover, the relation

(pq) (rs) = (p + r, q + s) + (p

+ ( p + *, qr) + (q + s, pr) + (pqrs)

leads, after throwing out a power of bt to precisely the same relation.

Art. 113. This remarkable circumstance greatly limits the number of operator
relations obtainable. It should be observed that any operator relation may be
multiplied throughout by any power of b and may be then used to obtain relations
between the numbers XT

VOL. CCVII. — A.

130 MAJOK P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS,

but no essentially new relations are thus obtainable ; for take a simple case

leading to

true for all valiies of m.

Ifwetake
we are led to

2 (NM-Nm_1)1c,8c,+i = (N.-
and if the former relation be written

the latter is merely f(m)_f(m_l} = 0 .

and further multiplication by b leads to the series of which the general term is

so that no new information is obtained.

Art. 114. The operator relation of the form D,Dt = a linear function of

7)
u

is not difficult to obtain.
I find that

and thence the formula for

D,D,DU

follows by taking DB as the operand on each side and then reducing the products

D,DM, D,+1D,,, D.+3DB)...
by the formula for D,De.

I find that

-('

U\ /t\ s+i\ fu\

'«+»

MAJOR P. A. MA.MAHON ON THE COMPOSITIONS OF M-.MUKKS.

131

and, generally, the product of any number of operators is expressible in the required
linear form.

Art. 115. With the object of connecting this theory of the numbers NM with that
of the n motors N (><l><\..), the generating function

will now to expanded in ascending powers of X, the coefficients of X toing functions
of the homogeneous product sums

The point of departure is the elementary formula

1 — Oi +
Remarking that

l

I write

equivalent to writing

and

then

1 — Oi + Oj— Oj-r ...

—na+... = (1— a,)(l—

/, xx r
(l-X)a. for a.,

/. . x, e
(1— X)'«. for a.,

(l-X)'A. for h.-,

and, as tofore, write
so that

whence, solving for A,
and

= u suppose ;

(1— X)aa+(l— X)-a3— ... = A;
.

u =

A=

i _i
1-X u '

where - — r- is the generating function under consideration.

1 — A.

Wnte Ht=hl+h,+h3+ht+...

H, = h3

H4 = A4

a 2

MAJOR P. A. MACMAHON ON THE COMPOSITIONS <>K NTMUKK'S.
6othat

therefore 1-X^ = 1_XHl+X«Hf-X'Ha+ ..

I ~~ A

and thence 1 _1 H!— XH3 + X2H3— ...

— ^_ j_ j

-XH1 + X2H2-X3H3+.
Now let functions A A A A

-"•1 > -"-2 > -"-3 > -"-4 i • • •

be connected with u tr TT

til, ±12, tl3, 114, ...

in the same way that

«!, a2, a3, a4, ...

are connected with , , , ,

«ll %, «3, «4, •••

so that A TT

A! = M! ,

A2 = Hia— H2,
A3=H13-2H1H2+H3,

then I __

1 \rt I \2tT \3TT

1 — Atli + A. rl2 — A H3+ ...

and — = l + (H1-XH2+X2H3-...)(l+XA1

1 ~~ J\.

On the dexter the co-factor of X' is

which has the value

Since «^-h>

is a well-known identity in the elementary theory of symmetric functions. Hence

-L- = 1 + A, + XA,+XSA3+X3A4+...;
1— A

or, as we may write it,

1

, + XA2-|-X2A3+X3A4+..M

+ X:'(H14-3lVILfH/+2H,H3-H4)+...

MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBKKR. 133

Art. 116. The preceding pages show thut the coefficient of

A

in the expansion of

-X) '<a-(l -
is equal to

the summation heing for every partition

(pqr...):'
and this, from the theory of the numhers

is equal to

SS...S/

PI PI P-

the summation being for all integer values of

or, the same thing, for the compositions of all numhers into exactly m parts.

Hence

SAfc-A,-Ht<

= A3=HI11=H1s-2H,Ha+Hs,

a remarkable result.
Art. 117. Since

we find, putting X = 1,
and, since

134 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.

and thence, as an easy deduction,

(for observe that, for the operand - , D.'E.sID,), and thence, by an easy step,

\ 1 ~~ T| /

[ 135 ]

TIT. On the Refractive Indices of Gaseous Potassium, Zinc, Cadmium, Mercury,

Arsenic, Selenium and Tellurium.

liy C. CUTHBERTSOX and E. PARR METOAI.FK, H.Sc.
Communicntfd by Professor F. T. TROUTOX, F. It.S.

Received October 23,— Read November 18, 1906.

WE have continued, with more perfect apparatus, the enquiry recorded in a previous
paper* on the refractive indices of the vapours of elements which are not gaseous at
ordinary temperatures. The instrument employed was the refractometer of JAMIN,
and the arrangement of the apparatus has been fully described in the paper just
cited, so that it requires only a brief recapitulation here.

Two similar exhausted tubes are placed in the paths of the rays of monochromatic
light between the mirrors, and a known weight of the element is vaporised in one of
them, of which tlie volume is known. The tubes are heated by means of a simple
form of electric furnace, consisting of two coils of nickel wire embedded in asbestos.

The number of interference bands which pass a fiducial mark in the field of the
observing telescope during the evaporation, or return during the condensation,
combined with the other data, give the index for the wave-length employed.

The calculation is as follows : —

If N be the number of bands observed to pass the fiducial mark, X the wave-length
X the length of the tube, in the refractive index observed, and /i the refractive index
at the standard density selected, we have

and

p.— 1 _ standard density
7j> — 1 observed density

_ '00009 x atomic weight_qf element volume of tube

atomic weight of hydrogen weight of element volatilised

Hence

,\ _ NX '00009 x atomic weight of element x volume of tube
X atomic weight of hydrogen x weight volatilised

* 'Phil. Trans.,' A, vol. 204, p. 323, 1905.
VOL. OCVII. — A 415. j.07

136 MESSES. C. CUTHBERTSON AND E. PAKR METCALFE

The standard density selected is that in which the number of atoms of the element
per unit volume is equal to the number of atoms contained in unit volume of hydrogen
at 0° C. and 760 millims.

In the present work the original procedure underwent several important improve-
ments. Repeated failure had shown that the glass tubes formerly used, with plate-
glass ends ground in and luted with shellac, were useless above 280° C., when the
shellac charred and the tubes cracked.

Recourse was had, therefore, to tubes of fused silica, which were made with
admirable skill by the firm of HERAECS, of Hanau. The ends of the tubes, through
which the interfering rays passed, were plates of the same material ground optically
flat, fused into the tubes and again polished, so that the whole formed an air-tight,
homogeneous enclosure, which could be heated to the highest temperature employed
(about 850° C.) without fear of softening, and could be heated and cooled locally with
great rapidity without cracking.

This property of the silica permitted the introduction of a second improvement in
the procedure. In the earlier work the two sections of the furnace were joined up in
the centre, and the whole length of the tubes was heated in one operation, so that the
observer had sometimes to remain with his eye at the telescope for several hours
while the furnace heated and cooled. This tedious method was now abolished. The
two sections of the furnace were separated by a gap of about an inch, and their inner
ends covered with thin asbestos boards. In these boards holes were punched, through
which the silica tubes passed. In this way the greater part of the tubes could be
raised to a high temperature while the gap in the middle remained comparatively
cool. With the object of obtaining a reasonable equality in the temperature of
the two halves of the furnace, the sections were made in all respects as nearly
similar as possible, and the heating coils were connected in parallel to the lighting
circuit.

When the furnace had reached the temperature which was found by experiment to
be more than sufficiently high to evaporate the whole of the charge employed, cold
water was dropped on the exposed part of the tube until it was certain that the
temperature there must be below 100° C., and, consequently, that there could be
practically none of the element in a state of vapour. The observer then noted the
position of the bands in the telescope, in relation to a pointer fixed on the further
mirror, and his colleague quickly heated the central portion of the tube with a Bunsen
flame. The bands now moved rapidly and attained their maximum in a few minutes.
The heating was continued for about a minute after the stopping of the bauds
indicated the complete vaporisation of the charge ; the flame being then removed, a
second reading was obtained as the element condensed. When nearly all the bands
had passed, water was again dropped on the tube till it was certain that the zero had
been reached. Meanwhile the temperature of the rest of the furnace was kept
approximately constant, and thus many minor sources of error due to inequality of

ON THE REFRACTIVE INDICES OF GASEOUS POTASSIUM, ETC. 137

the length of the tubes, or of the thickness of the ends, or to unequal heating of the
air, which had given trouble previously, were avoided.

In the case of zinc, the temperature necessary was so high that the glass
diaphragms used to close the outer ends of the furnace began to soften and buckle.
This difficulty was overcome by substituting worked plates of fused silica.

In the course of the work it was found that the dispersion would be considerable.
Arrangements were therefore made for determining the index for more than one wave-
length. A mercury vapour lamp of BASTIAN'S pattern was found to give an excellent
green (X = 4460), and red light (X = 6562) was obtained from a hydrogen vacuum
tube. But these methods proved unsatisfactory, and eventually it was found possible
to work with approximately monochromatic light sifted out with a slit from the
spectrum of the white light of a Nernst filament dispersed through four glass prisms.
The light thus obtained was sufficiently monochromatic to give twenty or thirty sharp
black interference Iwinds, and it was used in all the later experiments.

Potassium.

Of the elements whose indices still remain to be measured in the gaseous state, the
most important group is that of the alkalis. It was, therefore, decided to begin with
potassium, which seemed likely to prove the easiest.

So far as we are aware, only one attempt has hitherto been made to measure the
index of a member of this group. In the course of his brilliant research on the
optical properties of sodium vapour, Professor R. W. WOOD* measured the retardation
of light in passing through a column of dense vapour, and compared its value at
different points of the spectrum with that produced at the wave-length of the yellow
helium line. By the application of the Sellmeier dispersion formula to his results he
deduces an index of T0000275 for infinitely long waves for sodium vapour saturated
at 644° C. Unfortunately the density of the vapour corresponding to this tempera-
ture is yet unknown, so that it is impossible from the data available to calculate the
al)solute atomic refractivity of sodium.

The relative values for different wave-lengths are, of course, independent of the
density, and in themselves constitute a most interesting exemplification of the
variation of the index in the neighbourhood of an absorption band. Thus, the index
increases from the infra-red to the line D,, where it Incomes very large. On the blue
side of D2 the index increases rapidly from very small values, but remains less than
unity even as far as X = 2260.

In view of these facts the investigation of the index of potassium seemed likely to
yield results of similar interest. But, unfortunately, the chemical difficulties proved
so great that, in spite of numerous attempts, it was found impossible to obtain
absolute values with the apparatus employed. At the temperature at which

* ' PhiL Mag.' September, 1904.
VOL. OCVII. — A. T

138 MESSRS. C. CUTHBERTSON AND E. PARR METCALFK

potassium evaporates, it attacks lx)th glass and silica, so that it is not possible to
obtain, in vessels of these materials, a density of the vapour sufficient for a quanti-
tative determination of the absolute index. Eventually, however, by adopting the
device of heating all but a small portion of the refractometer tube to a sufficient
temperature, and then rapidly heating the cool part, a sufficient density of vapour
was attained to permit of some qualitative observations.

Since reliable numerical results could not be looked for, no attempt was made to
purify the potassium beyond distilling it in vacua after prolonged heating to expel
occluded hydrogen. The distillate was run into capillary tubes, which were sealed
off in short sections, and, by a device which need not be described, the operations of
breaking the capillary and introducing it into the refractometer tube were performed
entirely in vacuo.

Experiments were, at first, made with sodium light, but it was found that at a
temperature of about 220° C. the interference bands disappeared, though the light
was not wholly absorbed by the vapour. This effect might be accounted for by
supposing that the potassium contained a small quantity of sodium, or that sodium
was set free by the action of the potassium on the glass tube, for WOOD has shown
that the great dispersion of the vapour of sodium in the neighbourhood of the D
lines has the effect of destroying interference bands formed by light of those wave-
lengths, when even small quantities of vapour are present.

The sodium flame was therefore replaced by a Bastian mercury lamp which gave
good bands with the wave-length 5460. Several experiments were made with this
light with refractometer tubes both of glass and of silica ; and it was found that
the evaporation of the potassium was attended by a movement of the bands in the
direction corresponding to a refractive index less than unity. On one occasion no less
than four bands were observed to pass during the heating and to return during the
cooling.

These experiments at first suggested that the quantity of sodium present was
greater than had been suspected, and that its influence on the index dominated that
of the potassium. If this were so we should find that an experiment with light of
greater wave-length than that of the D lines would give a very high refractive index.
In order to test this surmise, experiments were made with the red light supplied by a
hydrogen vacuum tube whose Ha line was so strong as to give good interference bands.
But in this case also the bands moved in the same direction as those formed by the
light of the mercury lamp. We are, therefore, driven to the conclusion that the
index of potassium is less than unity both for X = 5460 and for X = 6562.

These results showed that no experiments on the indices of the alkali elements
were likely to repay the labour involved until the chemical difficulties had been
overcome and arrangements could be made for obtaining the index for very long
waves. It was, therefore, determined to abandon the attempt for the present, and to
turn to some other series.

ON THE REFRACTIVE INDICES OF GASEOUS POTASSIUM, ETC. ISO

Cadmium. (Atomic Weight 112.)

For experiments with this element the purest metal to be obtained from KAHI.BAUM
used. The arrangement described alxwe acted very well. At a temperature
l>etween 600° C. and 700° C. enough cadmium evaporated to give a shift of 20 or
30 bands. There was no marked absorption. The principal difficulty in obtaining
concordant results lay in the reading of the bands, which were unsteady owing to the
mirage caused by currents of unequally heated air, and in many cases were badly
illuminated. It was never possible to read to less than a quarter of a band ; and if
the conditions were bad, it was exceedingly easy to miss a whole bund, or even two,
or to count one twice. It was for this reason that the number of readings recorded
is so large.

Table I. exhibits the results obtained. Those readings which appear obviously
incorrect are enclosed in brackets. Four different charges of cadmium were used and
numerous readings were made with each. The results obtained with the second
charge may be neglected. In this case the bands were observed by the old method,
while the whole furnace was heated in a single operation, and the correction for
"end effects" was found to l)e of the order of 25 per cent, of the reading, and was
not trustworthy. The other three charges gave fairly consistent results, as will be
seen from the summary.

Zinc. (Atomic Weight 65.)

The next element dealt with was zinc, in the hope that the completion of the series
zinc, cadmium, mercury might afford useful information. Again, in this case, the
pure metal supplied by KAHLBAUM was used. Only two charges of zinc were used,
and much trouble was experienced in overcoming the effects of air convection currents.
No absorption l>and was oliserved. It is to be regretted that the number of bands
read was so small ; but the furnace was not adapted for reaching a higher temperature
than about 850° C., and it was not possible to evaporate a larger quantity at this
temperature.

Table II. exhibits the results.

Arsenic. (Atomic Weight 75.)

Arsenic, free from iron, was obtained from two sources, and a series of determinations
was made with each specimen.

This element proved the easiest of all those attempted, and concordant results were
quickly obtained (see Table III.).

Selenium. (Atomic Weight 79.)

Experiments were made with two charges of this element. In the first series the
readings were somewhat uncertain, and the result may be neglected. The second

T 2

140

MESSRS. C. CUTHBERTSON AND E. PARR METCALFE

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series gave values consistent with each other. The vapour exhibits a strong absorption
band extending fmni the violet so far towards the red that no readings could be made
at, the wave-length 5183, as, when five or six bands had passed, the light was

Oninplrtcly absorbed (Srr Table IV.).

Tellurium. (Atomic Weight 128.)

The specimen used was obtained from KAHLBAUM and, for the first experiment,
was redistilled. In this case also there is a strong absorption band in the violet,
which pi-evented readings being made beyond X = 5460. Tellurium was found to
evaporate with difficulty at a temperature hardly lower than that required for zinc,
and it was necessary to use very small quantities, so that the number of bands
observed was small, and the values obtained possibly less accurate. Two charges
were used. In the first day's work with the second charge the temperature was not
taken sufficiently high, with the result that the band readings are about G per cent,
too low ; but they are nevertheless recorded, as they show relative numbers for the
dispersion effect which are useful in corroborating the other results (see Table V.).

Mercury. (Atomic Weight 200.)

This was the first element dealt with by C. CUTHBERTSON three years ago. As a
check on the older method of working, another measurement of the refractivity of
mercury vapour was made by E. P. METCALFE, the silica refractometer tubes and the
divided furnace being now used. One charge only was employed, and observations
were taken with light of four different wave-lengths. The value of the refractivity
now obtained for D light (1866) is in good agreement with the previous deter-
mination (1857). The accuracy of the band readings seems to 1», for the I) line, to
within alxiut £ per cent., for the other wave-lengths to within about 1 per cent, (see
Table VI.).*

REMARKS.
Dispersion.

The present research was designed to obtain a single value for each element, and it
was only with the object of satisfying ourselves that the refractivity measured was
not affected to an abnormal degree by the existence of an absorption band for a wave-
length near to that by which the observations were made that readings were taken
with different wave-lengths. Each band reading being completely independent of
the rest, and the degree of accuracy being, apparently, not much greater than to

[* Subsequent work by Mr. METCAIJE renders it probable that the true value for A = 6562 is 1} to 2
per cent, higher than that given in Table VI. — January 24, 1907.]

144

MKSSRS. C. CUTHBERTSON AND E. PARR METCAI.FK

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ON THE REFRACTIVE INDICES OF CASEOUS POTASSIUM, ETC.

145

within ali. MII 2 per cent., an accurate record of the dispersion was hardly to be
expected. We have not overlooked the importance of determining the dispersion in
every case through a wide range of wave-lengths ; but the difficulty of obtaining
values sufficiently accurate to yield useful information when introduced into a
dispersion formula was found to be insurmountable with the present apparatus in the
time at our disposal. We hope, however, in the future, to make a further attempt to
obtain the dispersion with improved apparatus.

The results appear interesting. In every case, except that of potassium, the
dispersion observed is normal in direction.* But its magnitude is very great. Thus,
in the case of tellurium, the refractivity varies from about 2370 for X = 6562 to 2620
for X = 5460, or nearly 10 per cent. For zinc the variation is about 7 per cent, for
the same range.

Refraction and Refraction Equivalents.

It is probable that the refractive indices for infinitely long waves would be
considerably smaller than those here observed. But even after making the most
liberal deduction, it is certain that the indices of zinc, cadmium, and tellurium are
higher than the highest (viz., that of iodine) of which we had previous knowledge.
It is interesting to compare them with the refraction equivalents of GLADSTONK and
others. This is done in Table VII.

TABLE VII.

Refraction equivalents.

Refmctivities now found (gaseous).

GraitsroxK.

HAAGK.\.§

KANNOXIKOFF.II

(p. - 1) 10".

Early.t

Later.J

R..

R.v.

R..

Rv

X
6562.

X
5893.

X
5460.

X
5183.

I

As .

15-4
10-2
13-6

1-3

15-4
9-8
13-1
30-5

20-22
19-89

-

18-84
18-64

9-8
13-03

9-4
12-66

1960
2675
1530

1550
2060
2675

1 :.,;-.

139

1580
2150
2725
1570

2070

•_;-«

Zn . .

Cd
Se . .

H

h The apparent exceptions to this statement are attributable to errors of observation,
t 'Journal Chem. Soc.,' 1870, p. 101 ; 'Phil. Trans.,' 1870, p. 9.
J 'American Journal of Science,' 3, 29, 1885, p. 57.
§ 'Pogg. Ann.,' 131, 1867, p. 125.
|| 'Journal fur Praktische Chemie,' 31, 1885, p. 339.
VOL. CO VII. — A. v

MESSRS. C. CUTHBERTSON AND E. PARR METCALFK

The results are very curious. GLADSTONE'S refraction equivalent for arsenic* is
about twelve times that for hydrogen, while the gaseous refractivity is eleven times
that of hydrogen. But the values of GLADSTONE for zinc and cadmium are just half
those now obtained for the gaseous state. HAAGEN, however, has a value for zinc
which corresponds fairly well with the refractivity. Finally, GLADSTONE'S value of
selenium is about double that shown by the refractivity.

Relative Refi-actimties of the Elements.

Previous work in this field had brought to light certain relations between the
refractivities of the elements which seemed too regular to be due to chance, and the
present work was undertaken mainly with the object of ascertaining whether similar
relations existed in the case of other elements.

The results have been partly successful and parti}' unsuccessful. It was at once
manifest that, in view of the great dispersion and in the absence of trustworthy
values for infinitely long wave-lengths, the search for simple integral ratios between
the refractivities must be postponed. But in one group of elements traces of a
tendency to conform to such a rule may, we think, be fairly claimed.

Table VIII. shows the elements arranged in the order of their atomic weights, with
the refractivities appended in those cases in which they have been measured in the
gaseous state for the D line. The form of the table is suggested by the ratios
existing between the refractivities of some of the elements, and is, so far as we know,
somewhat different from previous forms of the periodic table.

Thus, since the refractivities of nitrogen, oxygen, fluorine and neon are respectively
one fourth of those of phosphorus, sulphur, chlorine and argon, it would seem that
each of these groups should form a horizontal row, and since the refraction equivalents
of potassium, rubidium and caesium are in the same ratio as those of argon, krypton,

* In this connection it is interesting to compare the values now found with those suggested by the
indices of some compounds of the elements.

As an instance we may take the case of arsenic trichloride, whose index (as found by HAAGEN) is
approximately 1'6. Converting this number into the corresponding one for a gas by the formula of

LORENZ, ^- = constant, we arrive at the figure 1920; of this 1152 may be subtracted for the

ft- + 2 a

chlorine atoms, leaving a Iwlance 768 for one atom of arsenic, or 1536 for two atoms of arsenic, a result
which corresponds well with the number 1550 now found. But the additive rule, as is well known, is not
of universal application. It fails conspicuously in the case of the fluorine compounds of the sulphur,
selenium, tellurium group. The present writers were, through the courtesy of Dr. E. B. R. PKIDEAVX,
afforded the opportunity of measuring the refractivities of these compounds. The results have been
published already in Dr. PRIDEAUX' paper in ' Trans. Chem. Soc.,' 1906, vol. 89, p. 330. For the
refractivities of the hexafluorides of sulphur, selenium, and tellurium we found the values 783, 895, and
991, the corresponding numbers deduced by the additive rule being 1116, 1356, and 1826 respectively.

ON TUB REFRACTIVE IN DICKS OF GASEOUS IfJTAKSIUM, ETC. 147

3

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148 ON THE REFRACTIVE INDICES OF GASEOUS POTASSIUM, ETC.

and xenon (i.e., as 2 : 3 : 5) the table has been rearranged so as to bring these together
in such a way that a single factor is characteristic of each row.*

From such a table we should expect the refractivities of selenium and tellurium to
be to that of sulphur as 3 and 5 respectively to 2. The values found are in both
cases rather low, but that for selenium is not notably so. In the other cases the
regularity breaks down. The value for arsenic is much nearer -f- that of phosphorus
than f. The values for zinc do not bear to those for cadmium the ratio 3 : 5, while
the refractivity of mercury is actually less than that of either, thus exhibiting the
only case yet observed of a gaseous refractivity which is lower than that of another
element of the same group and of lower atomic weight.

In short, it may be said that the rule of simple integral ratios between their
refractivities is probably confined to the elements lying near the centre of the table
which are shown enclosed within the heavy rectangular boundary. To these we
may possibly add the group potassium, rubidium, caesium, in which the refraction
equivalents seem to indicate conformity to the rule. Within the heavily marked
enclosure the regularity is very striking, except for the two, or possibly three,
elements in the lower left-hand corner.

A general survey of the table, with the new additions, confirms the view already
held that, in each group of elements, refractivity increases witli atomic weight. But
mercury, which stands in a lower horizontal row than the other elements examined,
forms a striking exception to this rule, and it remains to be seen whether this is
merely an isolated case, or whether it is characteristic of all the heaviest elements.
The refraction equivalents of GLADSTONE lend some colour to the conjecture that
there may be a falling off in refractive power when the atomic weight exceeds a
certain limit, but, in view of the discrepancies exhibited in Table VII., it would be
dangerous to place much reliance on their indications. It is also remarkable that in
each horizontal row of Table VIII. refractive power increases as we move to the left,
in spite of the decrease in atomic weight ; there can be little doubt, therefore, that it-
is intimately connected with the valency.

We have to express our cordial thanks to Professor TROUTON and the staff of the
Physical Laboratory at University College, London, for assistance and advice, and to
the Royal Society for a grant in aid of the research.

* This arrangement was first suggested by Professor A. W. POUTER. It will bo observed that it fits in
very well with the sequence of the atomic volumes.

[ 149 ]

IV. (hi the Discharge of Negative Electricity from Hot Calcium and from Lime.

By FRANK HORTON, D.Sc., B.A., Fellow of St. John'n College, and Clerk-
Maxicell Student of the University, Camlnidge.

Communicated by Professor J. J. THOMSON, F.R.S.

•

Received December 10, 1906,— Read January 31, 1907.

INTRODUCTION.

THE discharge of electricity from hot metals has been the subject of a great number
of researches by different authors. The metal chiefly used in these experiments has
been platinum, on account of its high melting-point, its stability in air, and the ease
with which it can be obtained in a state of purity. In the present experiments
calcium was chosen for investigation because of its strong electropositive character.
Since this implies a great attraction for positive electricity, it would l)e expected that
the negative corpuscles would escape more readily from calcium than from platinum.
It should, therefore, be possible to obtain a measurable " negative leak " from calcium
at a much lower temperature than from platinum or other less electropositive metal.

The first method of experimenting employed consisted in measuring the saturation
current from an electrically heated calcium wire to a surrounding electrode, both
being placed in a vacuum, but it was found to be impossible to get a clean surface of
calcium in this way, for the metal combines with the oxygen, nitrogen, and water
vapour in the air, and becomes more or less covered with a coating of calcium
compounds before the apparatus can be fitted up. Another difficulty was soon
discovered. When the wire was heated to above a dull red heat, the vapour pressure
of the metal was sufficient for it to volatilize and condense on the colder walls of the
tube. Thus the wire got thinner at its hottest point, and, consequently, the
temperature there rose, and the sublimation increased, until in a few seconds the wire
had broken through.

It was finally decided to make use of the volatility of calcium in order to obtain a
clean surface of the metal. The method of experiment was to fix up a platinum strip
as the cathode in a vacuum tul)e and to ascertain the manner in which the current
from this to the other electrode varied with the temperature of the strip, with the
difference of potential between the electrodes, and with the gas pressure in the

VOL. OOVTI.— A 416. 4-6-07

150 DR. FRANK HORTON ON THE DISCHARGE OF

apparatus. The platinum strip was then covered with a layer of calcium by meaus ot
sublimation, and the current between the two electrodes was measured again. It
might here be stated that a current between the two electrodes was observed only
when the platinum strip was used as the cathode, the positive leak being too small to
be detected by the galvanometer used. In what follows, therefore, the platinum strip
will be spoken of as the cathode.

When the observations of the negative leak from the calcium-covered cathode had
been made, the calcium was oxidised to lime, and the leak measured again. In this
way the negative leak from metallic calcium was compared with the negative leak
from the same amount of metal in the form of oxide. The full account of the
experiments and results is divided for convenience into the following sections :—

( 1 ) Description of the apparatus, &c. ;

(2) Investigation of the negative leak from platinum ;

(3) The negative leak from calcium in helium ;

(4) The negative leak from lime in helium and in hydrogen ;

(5) Summary of results, and conclusion.

(1) Description of the Apparatus, &c.

The glass apparatus shown in fig. 1 was found to be the most convenient form of
discharge tube for these experiments.

A is a platinum strip which can be heated electrically, the current being supplied
by means of the thick platinum leads G. The leak from this to the platinum
electrodes B, C, which together form the anode, was measured by means of a delicate
d'Arsonval galvanometer, giving a deflection of 1 millim. for a current of 7'19x 10~10
ampere. The calcium wire D, from which the metal is to be sublimed, is beneath the
platinum strip, and about two centimetres from it. It can be heated by an electric
current led in through the thick copper leads F, which enter the bulb through
sealing-wax joints. The apparatus is connected with the mercury pump McLeod
gauge and P2O5 drying bulb by the side tube shown in the figure. The platinum
strip was 3'5 centims. long and 2 millims. wide. The calcium wire was about
6 centims. long and 1 millim. in diameter.

Since calcium is readily attacked by all the more common gases, the experiments
had to be conducted in an atmosphere of argon or helium. Either of these gases
could be admitted into the apparatus through a side tube. For the purpose of
purifying the gas a small discharge tube was fitted on to the apparatus. In this the
cathode was an alloy of potassium and sodium, made by mixing the metals in atomic
proportions. When a discharge from an induction coil is sent through this tube the
alloy gradually absorbs any gas that may be present, except argon or helium. The
argon or helium in the apparatus could thus be purified by running the discharge

NKi: \TIVK l.l.KCrUCITY FlmM HOT CALCIUM AND FROM LIME.

151

tube until the pressure as indicated on the MacLeod gauge remained constant. In
the first exj>eriments the gas employed was argon, while in the later experiment*
helium gas was used. Tin- helium wa-s purchased out of a grant obtained from the
Royal Society.

The arrangement of apparatus used to measure the current from the surface of the
platinum i> indicated in fig. 2.

To Pump

Earth

Fig. 2.

A, A, platinum anodes ; C, platinum strip ;
H, H, leads of heating circuit; V, volt-
meter ; B, )>attery ; K, key ; G, galva-
nometer ; K, tin-foil fuse.

F
Fig. 1.

One end of the platinum strip C was connected to earth and also, through a fine
tin-foil fuse, R, to the negative pole of a battery of small storage cells, B. The
positive pole of this battery was connected to the anodes A, through the key K, and
the sensitive d'Arsonval galvanometer G, which was well insulated on paraffin
blocks and served to measure the current. The difference of potential between the
electrodes A and C was determined by means of the electrostatic voltmeter V.
H, H, are the leads for heating the platinum strip.

The temperature of the cathode was determined by means of a thermocouple which
was welded on to the strip at its middle. The wires forming the thermocouple were

152 DR. FRANK HORTON ON THE DISCHARGE OF

pure platinum, and platinum with 10 per cent, of rhodium. They were of very small
diameter (0'0025 centimetre), so that the temperature of the strip was not materially
lowered by the heat conducted along them. Each wire was sealed into a fine glass
tube, and these tubes were sealed into the bulb vertically above the platinum strip at
E (fig. 1). The other ends of the thermocouple wires were some 20 centims. away
and were soldered on to wires from a d'Arsonval galvanometer, the junctions being
enclosed in a water jacket, through which a stream of water, at a known constant
temperature, was kept circulating. The thermocouple was standardised before the
platinum strip was placed in the bulb. Very small grains of pure potassium sulphate
were placed on the strip near to the junction. The strip was then heated by an
electric current, which was increased until the grains of sulphate (observed through
a microscope) just melted. The galvanometer deflection corresponding to this
temperature" was noted. Use was then made of the curve given by CALLENDAR*
for transposing the galvanometer readings into degrees Centigrade.

(2) Investigation of the Negative Leak from Platinum.

Professor H. A. WILSON has shown that, in order to get constant values of the
negative leak from platinum, great care must be taken to remove all traces of
hydrogen, as this gas has a huge effect on the negative leak. For this purpose
Professor WILSON recommends boiling the platinum in nitric acid. This method was
adopted in the present research. The apparatus shown in fig. 1 consists of two parts,
the lower of which, holding the calcium wire and leads, slides into the upper part and
is held in position by a sealing-wax joint. When the apparatus had been made, the
upper part containing the platinum strip to be used as a cathode was inverted and
filled with strong nitric acid. This was boiled for about an hour, and then the acid
was replaced by a fresh supply, which was also boiled for some time. After several
boilings with strong nitric acid the apparatus was washed out with distilled water
and dried by sucking dry air, filtered by passing through cotton wool, through it by
means of a water pump. The calcium wire was carefully cleaned with fine emery
paper and quickly sealed in position. The whole apparatus was then fixed on to the
mercury pump, and the air pumped out until a good vacuum was obtained. The
platinum strip was raised to incandescence by means of the current from 10 E.P.S.
motor cells. After the platinum strip had been treated with nitric acid there was
found to be only a slight increase of gas pressure inside the apparatus when the strip
was left at a high temperature for a long time. The slight evolution of gas is
probably hydrogen, which had been occluded in the platinum and was not completely
eliminated by boiling with nitric acid. The negative leak from the platinum was
found to decrease slightly as the gas was evolved, but it soon became fairly constant.

* CAU.ENDAR, 'Phil. Mag.,' vol. 48, p. 519.

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME. 153

In some cases the attainment of the constant state was hastened by repeatedly
washing the apparatus out with dry air and heating the platinum in that gas.
When the pressure did not increase on heating the platinum, the apparatus was
pumped down to as low a pressure as possible, and helium was let in to a pressure of
3 or 4 millims. The sodium-potassium alloy was then let into the special discharge
tube through a well fitting tap, and the helium purified in the manner already
described.

The relation between the current and the electromotive force was first investigated.
The results obtained were similar to those found by other observers for the negative
leak in air, oxygen, or nitrogen. For instance, at a pressure of O'OOS mi Him. the
current was saturated with a potential difference of 30 volts, the distance between
the electrodes being about 2 centims. With higher pressures of gas in the apparatus
the potential difference required to saturate the current was much greater, on account
of the formation of new ions by collisions with the gas molecules. With pressures
above about O'Ol millim. the current never became saturated, but increased more
and more rapidly with the potential as the latter was raised.

Mention must be made of a curious increase in the negative leak which was
obtained whenever the cathode was allowed to remain for some time in a good
vacuum. For instance, if the apparatus was left at a low pressure (0'005 millim.)
over night, the leak was always found to be much larger when tested on the following
day. This increase was sometimes as much as a thousand times the normal current.
It gradually died away when the cathode was left at a bright red heat for some time.
On investigating this effect it was found that the increased leak was connected with
the appearance of a dark substance on the surface of the platinum strip. The amount
of this was very small, and it was only visible when the platinum was heated. It
disappeared on long continued heating, and its disappearance was always accompanied
by a huge decrease in the negative leak. This phenomenon only occurred at low
pressures. When the apparatus was left at a pressure of 2 or 3 millims., the negative
leak was found to remain practically constant. An effect similar to this seems to
have been obtained by Professor H. A. WILSON in his experiments on the discharge
of electricity from hot platinum. He says* : " If the wire is simply left standing in
air at a low pressure for a long time, the leak is often greater than before on again
heating the wire." I think that the black stuff which could be seen on the platinum
strip in my apparatus must have been some compound formed by the action of the
mercury vapour upon the platinum. WILSON found that mercury vapour increases
the negative leak very considerably at high temperatures. This seems to indicate
that there is some action between the two metals.

The leak would sometimes increase enormously when the apparatus was left at a
low pressure for only a few minutes ; sometimes even in the course of an observation
it would increase to ten or twenty times the normal value. In one case a platinum

* H. A. WILSON, ' Phil. Trans.,' A, vol. 202, p. 243, 1903.
VOL. CCVII. — A. X

154 DR. FRANK HORTON ON THE DISCHARGE OF

strip gave a leak of 376 x 10~9 ampere with a potential difference of 40 volts at a
temperature of 1480° C. and pressure 0'0042 millim. of mercury. On testing again, two
hours later, at the same temperature and pressure, the leak was 5'05 x 10"' ampere !

This increase of the negative leak on standing at low pressures rendered it necessary
to have a pressure of several millimetres when comparing the negative leak before and
after subliming the calcium on to the cathode. Some observations were therefore
made to ascertain the manner in which the leak from the hot platinum varies with
the gas pressure in the apparatus over the range of pressures likely to be used in the
subsequent experiments. It was found that with 40 volts difference of potential
between the electrodes the negative leak at a constant temperature was nearly
independent of the gas pressure between 7 '5 millims. and 3 millims. If the pressure
was reduced below this, the leak decreased gradually until a very low pressure was
reached, when it suddenly increased again to many times its former value. This
increase is probably due to the cause mentioned above. It did not always occur at
exactly the same pressure, but generally at pressures below O'l millim. Sometimes it
only appeared after allowing the apparatus to remain at a low pressure for several
hours.

The conclusion from these experiments is that, for the purpose of comparing the
negative leak from platinum with that from calcium or lime, it is best to work with a
gas pressure of a few millimetres of mercury and to use a constant voltage of 40 volts,
for the current never becomes saturated at this pressure. Working with a constant
voltage comes to practically the same thing as measuring the saturation current in
each case, for the current passing under a constant electromotive force should be
proportional to the number of ions liberated at the surface of the cathode.

The following table contains the values of the negative leak from the platinum strip
at different temperatures in helium at a pressure of 3'236 millims. with a potential
difference of 40 volts between the electrodes. The series of observations was repeated
several times during the course of two or three days, and the values given were found
to be practically constant. The numbers in brackets refer to the order in which the
measurements were made.

TABLE I. — Negative Leak from Platinum in Helium Gas at a Pressure of

3-236 millims.

Temperature, Negative leak per

° C. centimetre (ampere).

1331 1- 95x10-"

1468 3-96 x 10-"

1542 1- 75x10-*

1571 2-99 xlO'7

1610 5-91 x 10~7

The values given above are slightly smaller than those given by H. A. WILSON for
the negative leak per square centimetre from platinum in air at a low pressure.

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME.

155

Professor WILSON gives for the leak at 1545° C. the value 6'38 x 10~7 ampere. The

smallest value I ever obtained for the negative leak in helium was 6'91 x 10~* ampere

per square centimetre of platinum surface at

1 540° C. By taking great precautions in cleaning

his platinum wires and purifying the air in the

apparatus, with a special view to getting rid of

all traces of hydrogen evolved by the wire when

heated, Professor WILSON was able to reduce

the leak to 10~* ampere per square centimetre

of surface at 1616° C. The values given in

Table I. for the negative leaks in helium at

different temperatures could be reduced to about

one-tenth by reducing the pressure of the gas in

the apparatus. Since Professor WILSON'S results

were obtained in a good vacuum, it seems that

the value of the negative leak in helium is

practically the same as in air under similar con-

ditions of temperature and pressure.

From the numbers in Table I. the curve in fig. 3 was drawn. It is similar to the
current-temperature curves obtained by other observers in air.

O. W. RICHARDSON and H. A. WILSON have found that the variation of the negative
leak with temperature can be expressed by an equation of the type

tempera.tune centigrade
Fig. 3. Relation between negative leak
and temperature of platinum cathode in
helium gas at 3 • 236 millims. pressure.

x =

where x is the current in amperes, 0 the absolute temperature, and Q and A are
constants. Q is a measure of the work done by a corpuscle in escaping from the
surface of the metal ; it can be obtained from any two values of x by means of the
equation

0_o

By the use of this formula the following values of Q were obtained from the observed
currents. The numbers in brackets refer to the observations in Table L, which were
used in calculating the value of Q.

TABLE II. — Values of the Constant Q calculated from Observations of Table I.

From oltservations

Mean temperature,

1400
1505
1557
1591

Q

(calories).

121,100 -|
lL':.,100 L,
121,800 f Mean
1 19,500 J

121,900.

X 2

156 DR. FRANK HORTON ON THE DISCHARGE OF

The variations in Q are not greater than can be accounted for by the errors of
experiment. The mean value is lower than that given by WILSON (viz., 131,100), but
this is probably due to the fact that the present experiments were performed with a
gas pressure of several millimetres in the apparatus, whereas WILSON'S result was
obtained in a good vacuum.

Taking Q = 121,900 and the current per square centimetre at 1610° C. as
5 '91 x 10~7 ampere, the value of the constant A in the formula x = A^e"^2* is
1'55 x 10', so that the equation for the current x (amperes) at the absolute
temperature 6 becomes

The following is a comparison of the currents calculated by means of this equation
and those found experimentally.

TABLE III.

Temperature, Negative leak in amperes per square centimetre.

°C. Observed. • Calculated.

1331 l-95x!0-» 1- 96xlO-9

1468 3-96xlO-« 4-06x10-"

1542 1- 75xlO-7 -l-73x!0-T

1571 2-99x10-* 2-95x10-'

1610 5-91 xlO-7 5-91 xlO'7

The observed and calculated values of the current agree very well, showing that
the formula expresses the experimental results with considerable accuracy.

(3) The Negative Leak from Calcium in Helium.

Having shown that the negative leak from the platinum strip was of the normal
amount, and that it varied with the temperature according to the established law,
calcium was sublimed on to it and the alteration of the leak caused thereby was
investigated. The sublimation of the calcium was performed by connecting the thick
copper leads of the calcium wire (F, fig. 1) to the alternating current from a
transformer and gradually decreasing the resistance in the circuit until the calcium
became red hot. It then sublimed, and the bulb was covered with a fine metallic
mirror, and the electrodes would be similarly covered with calcium. With practice it
was possible to regulate the current so that the wire did not fuse through on the first
heating. After observations of the negative leak had been taken, more calcium could
be sublimed on to the cathode, and the observations repeated. The gas pressure in
the apparatus increased during the process on account of the gas evolved by the
calcium. The discharge was therefore started in the potassium-sodium tube and kept
going until the whole of the evolved gases were absorbed by the alloy and the
apparatus contained only helium gas at the same pressure as before. In order to
see if the evolved gas increased the leak from the platinum strip, an experiment was

NEGATIVE ELECTRICITY PROM HOT CALCIUM AND FROM LIME.

157

made in which the calcium wire was warmed sufficiently to expel some gas from it,
but not to a high enough temperature to cause it to volatilize on to the platinum.
The evolved gas was then absorbed in the potassium-sodium alloy, and the leak from
the platinum strip in helium gas was again tested. It was found to be the same as
before the calcium wire had been heated.

The negative leak from the calcium-covered strip was found at several tempera-
tures. The observed values are given in the following table. The potential
difference between the electrodes was 40 volts, and the gas pressure, as before,
3-236 millims.

TABLE IV.

Temperature.

Current in amperes per square centimetre.

Q

(calories).

Observed.

Calculated.

840

3-92x10-*

3-44xlO-»

4-32x10-*

919

1-47x10-"

3-11 xlO-"

9-01 x 10-*

986

1-lSxlO-T

1-63x10--

5-01x10-*

1005

1-63x10-'

2-52xlO-T

7-67x10-*

1050

4-32x10-*

6-75xlO-T

9-33x10-*

1065

6-46xlO-7

9- 27xlO-7

_4

1117

2-36xlO-»

2-61xlO-«

9-11x10 «

9-62x10-*

1142

4-37xlO-«

4-19xlO-«

8-99x10-*

1220

2-36xlO-&

1-66x10-*

5-44x10-*

1238

2-95x10-*

2-23xlO-s

7-42x10-*

1310

9-22x10-*

1-47x10-*

4-32x10-*

1385

l-75xlO-«

1- 93x10-*

Mean value of Q = 7 -29 x 10~4.

The observed currents in the table are plotted against the corresponding temperatures
in fig. 4. The unit of current is successively multiplied by ten in passing to the right
from one curve to the next.

It will be seen that the curves obtained are exactly similar to those given by
platinum.*

The observations of the negative leak from calcium were made as quickly as

* See RICHARDSON, 'Phil. Trans.,' A, vol. 201, p. 497.

158

DR. FRANK HORTON ON THE DISCHARGE OF

possible, because the calcium gradually sublimed off the platinum strip if left heating
for a long time. It is somewhat surprising that it remained on long enough for the
above readings to be taken. The explanation is, probably, that the calcium melts

o i

fgd -Kg

800°^ goo5

Temperature centigrade

Fig-

Relation between negative leak and temperature of calcium cathode in helium
gas at 3 • 236 millims. pressure.

and combines with the platinum on the surface of the strip, to form an alloy from
which the calcium only slowly vapourises away. After long heating the whole of
the calcium could be driven off' the platinum, and the negative leak was reduced to
the value it had before the calcium was sublimed.

From the numbers given it will be seen that the leak from calcium is enormously
greater than that from platinum at the same temperature. For example, at 1385° C.
the leak from calcium is about 5000 times the leak from platinum, which means that
there are 5000 times as many corpuscles liberated per second per square centimetre
from calcium as from platinum at this temperature.

The values of the constant Q, deduced from successive pairs of these results, are
given in the last column of Table IV.

The large variations in Q are most probably due to the experimental difficulties of
measuring the negative leak from calcium. It was generally difficult to get a steady
reading of the current at any temperature, for the leak increased and decreased in a
capricious manner. Moreover, the series of observations had to be taken very quickly,
because the leak gradually decreased with continual heating, owing to the calcium
subliming away.

The mean value of the constant A, calculated from the temperatures and currents

NEGATIVE ELECTRICITY FROM WOT CALCIUM AND FROM LIME. 159

per square centimetre, given in Table IV. is 171 x 104, so that the equation for the
current per square centimetre from calcium at 0° (al**olute) becomes

The values of the negative leaks calculated from this formula are placed alongside
the observed values in Table IV.

The agreement between the observed and calculated currents is not nearly so good
as in the case of platinum. This is no doubt due to the difficulties attending the use
of calcium. In the course of the experiments a note was made in the laboratory
book that the currents recorded at 1050° C., 1065° C., and 1310° C. were steadily
increasing during the observations. We see from the table above that the observed
currents at these temperatures were considerably too low. If it had been practicable
to have left the apparatus for a few minutes before measuring the leak, the values
of the currents at these temperatures would, no doubt, have been nearer to the
theoretical value. The experiments, however, show in a quite satisfactory manner
that at a given temperature the rate of emission of negative corpuscles from calcium
is much greater than in the case of platinum ; and by applying the results to the
formula employed we see that this increased rate of emission is due to a decrease in
the value of the constant Q, that is, to a decrease in the energy required to set free
the corpuscles from the surface of the platinum ; for on any theory of the negative
leak Q is a measure of the work required to produce a gramme molecular weight of
corpuscles. The value found for Q for calcium (7 '29 x 104) lies between the value
found for platinum (T219 x 108) and that obtained by RICHARDSON for sodium
(6 '32 x 104). This is what one would expect, and indicates that the amount of
energy required to liberate the corpuscles is less the more electropositive the metal.

(4) The Negative Leak from Lime.

When the series of observations recorded in Table IV. had been made, the calcium
on the platinum strip was oxidised to lime by letting into the apparatus some pure
dry oxygen. The oxygen was prepared by the electrolysis of water and then passed
over sticks of caustic potash and some fused calcium chloride. It was let into the
apparatus to a pressure of 3 or 4 millims. On gradually raising the temperature of
the cathode the negative leak was at first only slightly greater than before the
oxygen was admitted, but soon it increased very rapidly as the calcium oxidised, and
a pale glow appeared in the discharge tube. After this luminous discharge had once
appeared the negative leak at all temperatures was much greater than before the
oxygen had been admitted into the apparatus. The following table gives the
negative leaks under a potential difference of 40 volts at various temperatures in
helium, the excess of oxygen having been absorbed by sending a discharge for some

160

DR. FRANK HORTON ON THE DISCHARGE OF

time through the tube containing the sodium-potassium alloy. The pressure of the
helium was, as in the case of the calcium cathode, 3 '236 millims.

TABLE V.

Temperature

Current in amperes per square centimetre.

Q

(calories).

Observed.

Calculated.

730

5-29x10-8

3-73x10-8

7-25 xlO4

785

3-55xlO-7

4-57x10-8

8-82 xlO4

820

l-37x!0-«

1-98x10-*

Il-19xl04

856

7-13xlO-«

8-14xlO-«

13-76X104

877

2- 19xlO-5

l-79x!0-5

9-55 xlO4

918

9-30xlO-5

7-61 x 10-5

6-59 xlO4

949

1-90x10-*

2-HxlO-1

9-91 xlO4

965

3-23xlO-4

3-57xlO-4

Mean value of Q = 9-58 x 10*.

The observed currents in the above table are plotted against the corresponding

temperatures in fig. 5. The curves obtained are of the
usual form for the negative leak from glowing solids.

By comparing Table V. with Table IV. it will be seen
that the negative leak from lime is enormously greater
than from metallic calcium under the same conditions,
the leak from a lime cathode at 950° C. being about the
same as the leak from calcium at 1400° C. This is
contrary to what we should expect on the supposition
that the negative leak is due to the escaping of the
corpuscles from the cathode, for the presence of an atom
of oxygen in the molecule of lime would hinder, by its
attraction for negative electricity, the escape of the
corpuscles, and we should expect, in consequence, that

the negative leak from lime would be less, under the
Fig. 5. Negative leak from lime ,... ,. , , AIJ.U

same conditions of temperature and pressure, than the
in helium at a pressure of .

3 -236 millims negative leak from the same amount of calcium in the

metallic state.

The values of the constant Q, deduced from successive pairs of observations, are
given in the last column of Table V.

700^ 800 900

Temperature centigrade

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME. 161

The variations in the value of Q are very considerable — nearly as large as in the
case of the calcium cathode. Several other sets of observations of the negative leak
from a lime-covered cathode were made, and in all cases there were considerable
variations in the values of the constant Q, calculated from the experimental numbers.
It should be mentioned that when the temperature was 877° C. a pale glow was observed
in the gas on one side of the cathode. This extended to both sides of the cathode on
raising the temperature to 918° C., and became brighter at each subsequent increase of
the temperature. The appearance of the glow was not marked by any abnormal increase
in the current, as will be seen by the perfect continuity of the curve in fig. 5. The
large values of Q obtained at this point seem to be purely accidental. From a large
number of experiments with lime cathodes the mean value of Q, calculated from
observations made just when the discharge became luminous, was not greater than
the average value for the whole series of observations. The appearance of the
luminous discharge will be treated of more fully in a later part of this paper.

The negative leak from the lime was tested again a few hours later. It was found
to have decreased very considerably, and the luminous discharge did not now appear
until the cathode had been raised to a much higher temperature than before. This
was found to be a general rule, namely, that long continued heating of the lime
diminished the negative leak. In the present case, after heating for about two hours
to 1000° C., the negative leak was only of about the same magnitude as in the case
of the calcium cathode. This will be seen by comparing the numbers in the following
table with those given for calcium in Table IV. :—

TABUS VI. — Negative leak from lime after heating to 1000° C. for two hours in
helium gas at a pressure of 3'236 millimetres.

Temperature, Negative leak, Temperature, Negative leak,

C. amperes per square centimetre. * C. amperes per square centimetre.

942 6-84xlO-» 1226 2-24 xlO'8

I"*1"' l-22x!0-7 1290 1 •81x10-'

1170 l-51x!0-« 1316 3-42xlO-«

The mean value of Q obtained from these observations is 1/34 x 10* — much greater
than the mean value given in Table V. The luminous discharge was not observed
in these experiments until the temperature was raised to 1316°C.

The mean value of the constant A calculated from the experimental numbers
tabulated in Table V. is 6 -42 x 10", so that the equation for the current per square
centimetre from lime at the absolute temperature 6 is

x = 6-42 x ioll0»e-»-««* '••/».

The currents calculated by means of this formula are given in the third column
of Table V.

From these figures it will be seen that the formula only roughly represents the
VOL. ccvn. — A. Y

162 DR. FRANK HORTON ON THE DISCHARGE OF

observed results. The discrepancies may be due to the unsteadiness of the negative
leak from lime which has been mentioned above. The leak was steadier than with
the calcium cathode, but not nearly so steady as with the glowing platinum. The
chief alteration of the leak was the gradual decrease as the heating of the cathode
was continued. This may have been due to a diminution of the amount of lime by
spluttering or by peeling off from the surface of the platinum, although no such
phenomena could be observed. It is not due to a decomposition of the lime by
electrolysis, for the author has shown* that no signs of electrolysis can be detected
when a current is sent through a vacuum tube from a lime cathode, and further
experiments with other lime cathodes showed that the negative leak decreased with
continual heating of the cathode, whether the discharge was passed or not. The
discrepancies between the observed and calculated values in Table V. follow from the
discordant values found for the constant Q.

It has already been mentioned that a large series of experiments with lime
cathodes all gave similar results. In order to see if these irregularities were peculiar
to the present method of experimenting, the values of Q were calculated from
WEHNELT'S values of the negative leak from lime given in the ' Philosophical Maga-
zine' for July, 1905, p. 87. The variations in the values of Q thus found were
somewhat greater than those shown in Table V. The mean value was Q = 50,900 —
considerably less than the value found in the present experiments. It thus seems
that the negative leak from lime is subject to irregular variations, and does not obey
the Wilson-Richardson law with anything like the accuracy of the leak from
platinum.

The fact that the negative leak from calcium is greater than from platinum at the
same temperature we have seen to be due to a decrease in the value of the constant Q,
that is, to a diminution of the energy required for the liberation of the corpuscles.
The value of Q for lime as found in the observations tabulated above is greater than
the value for calcium, and the fact that the current is greater in the case of lime
than in the case of calcium is due to the enormously greater value of the constant A
in the former case.

A theory to account for the negative leak from hot metals has been proposed by
RICHARDSON, t He supposes the negative leak to be due to the escape of the cor-
puscles which, on the ionic theory of metallic conduction, all conductors contain.
The corpuscles are supposed to move about freely inside the conductor, and to have
a distribution of velocities the same as the molecules of a gas. Corpuscles entering
the surface layers of the conductor with a normal velocity component greater than a
certain amount are supposed to escape into the surrounding space, and it is these
corpuscles which maintain the current forming the negative leak. From these
assumptions RICHARDSON has deduced a formula of the type x = A^e"^2*, and has

* 'Phil. Mag.,' April, 1906, p. 506.

t O. W. RICHARDSON, ' Phil. Trans.,' A, vol. 201, p. 497.

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME. 168

shown that the constant A is proportional to the number of free corpuscles in a cubic
centimetre of the conductor in question.

From experiments on the negative leak from hot platinum in air and in hydrogen
H. A. WILSON* has come to the conclusion that the phenomenon cannot be completely
explained by such a simple theory, and, moreover, he has shown that in order to
obtain the above formula it is not necessary to make any assumption as to the
manner of formation of the ions, but only to assume that they are produced in some
way at the surface of the hot platinum. In WILSON'S view, the constant Q is a
measure of the work required to produce a gramme molecular weight of ions at the
surface of the hot platinum, but experiments on the negative leak in hydrogen at
different pressures have led to the conclusion that the constant A cannot be regarded
as proportional to the number of corpuscles in a cubic centimetre of the cathode, and
WILSON shows that the number so deduced does not agree with the value found by
PATTERSONt from experiments on the variation of the resistance of platinum in a
magnetic field. A comparison of the values of A found for lime and for calcium in
the present research appears to support this view of WILSON'S, for it does not seem
possible that there can be 107 times as many free corpuscles in a cubic centimetre
of lime as in a cubic centimetre of calcium. Further, it is well known that the
electric conductivity of lime increases rapidly with rise of temperature. The author
has shown that this conductivity is mainly, if not entirely, metallic in nature, and, on
the ionic theory of metallic conduction, due to a large increase in the number of free
corpuscles contained in the substance, for it is improbable that the velocity of the
corpuscles increases to this extent. If, then, A is proportional to the number of
corpuscles per cubic centimetre, its value should increase with the temperature.
In the present experiments with lime, although there were considerable variations in
the value of A calculated from the negative leaks at different temperatures, there was
no sign of a progressive increase with increasing temperature.

From WEHNELT'S work on the discharge of negative ions from glowing lime and
other metallic oxides, RICHARDSON has drawn the conclusion that the corpuscles
proceed not from the glowing oxide, but from the platinum, and that the oxide
merely has the effect of lessening the amount of energy required to set them free.
This conclusion is arrived at from the fact that the number of corpuscles per cubic
centimetre calculated from the value of the constant A, as found from WEHNELT'S
numbers, is about the same as for platinum. The value of A for lime obtained in the
present experiments is much greater than the value obtained for platinum, the former
being 6'42xlOu and the latter 1 '55x10*. Other experiments with lime-covered
cathodes gave values ranging from A = l'23x 10l° to A = 7'12x 1013, the value given
by the observations recorded in Table VI., taken after the cathode had been heated
for a long time to a high temperature. The values of A, calculated from observations

• H. A. WILSON, ' Phil. Trans.,' A, vol. 202, p. 243.
t PATTERSON, 'Phil. Mag.,' 6, HI., 655.
Y 2

1(54 DR. FRANK HORTON ON THE DISCHARGE OF

with different cathodes, are thus seen to vary very considerably ; the values of Q, too,
were not in very good agreement. It seems likely that this may be due to the
platinum being more completely covered with lime in some cases than in others, for it
will be readily understood that the cathode could not be quite uniformly covered with
calcium at each attempt by the method of sublimation. In order to obtain more
accurate knowledge of the values of these constants, experiments must be made with
very carefully prepared lime cathodes. Meanwhile, the fact that a large emission of
negative corpuscles takes place from a Nernst filament at high temperatures seems
to indicate that in experiments with lime-covered cathodes the corpuscles proceed
from the oxide, and not from the platinum underneath.

In view of the experiments of Professor WILSON, which have shown that the
presence of hydrogen enormously increases the negative leak from platinum, it was
thought to be interesting to see if the leak from lime was increased by admitting
hydrogen into the apparatus. It was found that this is the case, the leak in
hydrogen being many times greater than in helium or oxygen. The effect of
introducing a little hydrogen into the apparatus is well shown by some observations
taken with a lime cathode which had been used for some days, and the negative leak
reduced to even a smaller amount than the values given in Table VI. The following
are the values of the negative leak from such a cathode. The gas present was a
mixture of helium and oxygen at a pressure of 3 '91 millims. The voltage used was
-40 volts.

TABLE VII.

Temperature, ° C. 1038 1382 1520

Negative leak in amperes 6 x lO"9 1 • 32 x lO"5 1 • 13 x 10~4

The above numbers are the smallest values of the negative leak at the temperatures
given that I ever obtained from a lime cathode. There was no sign of a luminous
discharge, even at the highest temperature. After taking these observations, some
pure dry hydrogen was let into the apparatus, and the cathode was gently warmed
until no further diminution of pressure took place. The resulting gas was a mixture
of helium and hydrogen at a pressure of 3 '81 millims. A luminous discharge was
now noticed when the temperature of the cathode was 1220° C. and the current
passing 3'12xlO~* ampere. The glow in the gas was of a pale blue colour, and
appeared only round the edges of the anodes. As the temperature of the cathode
was gradually raised and the current passing increased, the glow became more extensive
and brighter. At 1465° C. it was very white, and had gathered up into little balls
about points on the rim of the anodes. Although the temperature was raised as high
as was compatible with the safety of the cathode, and the current passing rose to one-
twentieth of an ampere, no cathode glow was obtained. The following is a selection
from a list of readings obtained in this series of experiments. The readings of the
negative leak, at temperatures over 1250° C., were taken with a milliainmeter. The

NKOATIVE KLHCTBICITY FROM HOT CALCIUM AND FROM LIME. 1«5

currents measured decreased rapidly as the heating of the cathode was continued,
especially at the highest temperatures.

TABLE VIII.

Temperature, Negative leak Temperature, Negative leak

C. in amperes. C. in amperes.

895 9-85xlO-» 1380 4-5xlQ-»

1013 5-69xlO-« 1465 1-4x10-*

1220 3-12xlO-« 1535 2'7xlO-»

1293 1-2 xlO-» 1620 4'7xlO-»

It is thus seen that the negative leak from lime is considerably increased by intro-
ducing hydrogen gas into the apparatus. H. A. WILSON has shown that hydrogen
greatly increases the negative leak from platinum, and has come to the conclusion
that the negative leak from platinum in air, or in a vacuum, is almost entirely due to
traces of hydrogen in the metal. WILSON reduced the leak to j5^066 part of that
ot«erved by RICHARDSON by taking precautions to remove such traces. It should be
mentioned that the currents in the above table are much larger than the negative
leaks from platinum in hydrogen obtained by WILSON, and the increase of current
cannot be merely due to the effect of the hydrogen upon the platinum.

In another experiment a lime-covered cathode in oxygen at a low pressure
(0'002 millim.) with a potential difference of 40 volts gave a negative leak of
4'6x 10~* ampere at 965° C. On pumping out the oxygen and letting hydrogen
into the apparatus and then pumping down to the same pressure as before, the
negative leak at temperatures below 900° C. was only slightly greater than the leak
at the same temperature before the hydrogen was admitted, but at 980° C. a faint
luminosity was seen in the gas round the cathode, and the negative leak increased
tvithout the temperature of the cathode being raised t>r the difference of potential
between the electrodes being altered. This increase was slow for a few minutes, but
afterwards became more rapid, and, although the temperature was lowered by
putting more resistance in the heating circuit, the negative leak increased to 60 milli-
amperes at 885° C. The luminous glow was then quite bright, and filled the whole
bulb. It was at first thought that this sudden increase in the negative leak was due
to the temperature of the cathode increasing while the discharge was passing, but
experiments showed that the temperature of the cathode went up only a few degrees
when the electric field was put on, and the leak gradually increased, even though the
temperature, as indicated by the thermo-juuction, was diminished by putting extra
resistance in the heating circuit.

On allowing the cathode to cool down, and then again testing at lower tempera-
tures, it was found that the negative leak at these lower temperatures was now much
greater than at the first observations. A measurable leak was obtained at a much
lower temperature than before, and the leak at 740° C. was 4'6x 10~* ampere — about
10* times as large as before the glow had been obtained in the discharge tube. On

166 DR. FRANK HORTON ON THE DISCHARGE OF

gradually increasing the temperature the luminous discharge began without any
sudden jump in negative leak taking place. I again found that after a certain
temperature had been attained and a large current was passing, I could decrease the
temperature of the cathode and still get the luminosity to continue and the current
to pass.

These experiments show that the negative leak from lime is enormously increased
by replacing the gas in the apparatus by hydrogen. With a lime cathode in hydrogen
at a pressure of O'Ol millim. I obtained a current of 0'15 ampere per square
centimetre at about 900° C., with a difference of potential of 40 volts between the
electrodes. This is the largest negative leak I have measured under this potential
difference.

The appearance of the luminous discharge is of great interest. Generally the
luminosity began round the cathode, and was of a very faint blue colour, getting
whiter and more extensive as the temperature of the cathode was increased. When
the cathode was unequally covered with lime, the discharge could be seen to radiate
out from a few points only. The appearance of the luminosity at any point depends
on the current density at that point, and with a very evenly covered cathode large
currents could be made to pass through the tube without any signs of a luminous
discharge appearing. The appearance of the luminosity also depends on the potential
difference between the electrodes. The luminosity could not be obtained with a
potential difference of less than 18 volts ; and it seems probable that this is the value
of the anode fall of potential, for the cathode fall is reduced to a very small amount
by the enormous emission of negatively charged corpuscles from the cathode.* As a
rule, the luminous discharge gradually became visible, and increased in brightness as
the temperature of the cathode was slowly raised. When the luminosity appeared
gradually there was no sudden jump in the current passing. This is well illustrated
in the curves of fig. 5. On the other hand, when the potential difference between
the electrodes was much greater than 40 volts, the luminosity usually appeared quite
suddenly and was accompanied by a sudden increase in the negative leak.

The current density obtained with a calcium-covered cathode was, generally, not
sufficient to produce a luminous discharge, but on one occasion, on heating the
cathode to a much higher temperature than usual, a faint luminosity was observed.
This was at about 1520° C., and the current passing through the tube was 4 milli-
amperes. Some interesting experiments were made with this cathode. The tempera-
ture was kept constant, and the potential difference between the electrodes was
increased from zero by two volts at a time. No luminosity was obtained until a
potential difference of 20 volts was reached. With this voltage a pale glow was seen
round the anode. This glow increased in brightness as the voltage was increased.
With 28 volts the glow left the anode and a pale luminosity appeared round the
cathode. At the same time the current passing increased from being too small to

* WEHNELT, 'Phil. Mag.,' 6, vol. 10, 1905.

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME. 167

measure on a milliammeter to 2'8 milli-amperes. These experiments were performed
in helium at a pressure of 3'28 millings. Similar results were obtained with lime
cathodes. For instance, during the observations recorded in Table VI I L a faint
luminosity was noticed round the edges of the anode when the cathode was at
1220° C. This luminosity increased in brightness as the temperature was raised, but
with a potential difference of 40 volts the glow was always on the anode only. It
was found that at 1410° C. it required a potential difference of 74 volts to give a
luminosity round the cathode. With 72 volts there was a very bright anode glow,
and the current passing was 4*5 milli-amperes. With 74 volts a brilliant cathode
glow was obtained, and the current suddenly increased to 0'5 ampere. Further
experiments showed that the difference of potential required to give the cathode glow
was less the higher the temperature of the cathode.

The appearance of the luminosity round the cathode has been studied by
Professor J. J. THOMSON,* who worked in a slightly different manner from that
described above. By keeping the temperature of the cathode constant, and very
gradually increasing the potential difference by means of a potential divider,
Professor THOMSON found that the luminosity always appeared quite abruptly and
was accompanied by a very rapid increase in the negative leak. In one case, at
1400° C., an increase in the potential difference of -j-J^ of a volt caused a bright
luminosity to appear and increased the current forty-fold. A result similar to this
was obtained in the experiments now recorded, when the cathode was at a high
temperature and the voltage gradually increased ; but if the temperature of the
cathode was not too high, a luminosity round the anode was first obtained, and this at
a certain potential difference appeared to leave the anodes and surrounded the cathode.
This inversion of the appearance of the discharge was accompanied by a large increase
in the current.

In the present experiments, when a difference of potential of 40 volts was used and
the temperature of the cathode gradually increased, the luminosity appeared sometimes
round the anodes and sometimes round the cathode, but always quite gradually, except
when a cathode newly covered with lime was used. In this case the luminosity did
not appear until a temperature higher than usual had been reached. Under these
circumstances it generally appeared quite suddenly and then increased in brightness,
although the temperature was kept constant or, in some cases, actually lowered.

Professor THOMSON has concluded from his experiments that the gas becomes
luminous in consequence of the internal energy of the atoms increasing, under the
bombardment of the corpuscles shot out by the cathode, to such an extent that the
equilibrium of the atomic system becomes unstable and an explosion occurs. This
explosion results in an expulsion of corpuscles and such a shaking up of those left in
the atom that they vibrate so vigorously that the energy radiated is sufficient to
produce luminosity. When the luminosity appears abruptly, we must imagine that
* Royal Institution Lecture, Friday, Jamiary 19, 1906.

168 DR. FRANK HORTON ON THE DISCHARGE OF

just before it occurs the atoms are in such a state that a small change in the electrical
conditions is sufficient to cause them to pass from a condition in which they are giving
out no light to one in which they are brightly luminous. Now the current passing
through the tube increases with the potential difference between the electrodes at a
rate which increases rapidly with the temperature of the cathode. The higher the
temperature of the cathode, therefore, the greater will be the effect of a given increase
in the potential difference, and whereas at low temperatures the appearance of the
luminosity may be quite gradual, the same increment in the potential difference may,
at high temperatures, make all the difference between no glow and a very bright one,
so that the luminosity appears quite suddenly. In the present experiments the gas
pressure was so great that the current was never saturated, but increased at an ever
increasing rate with the potential difference. Under these circumstances a similar
argument will apply to the case of the potential being kept steady and the temperature
being gradually increased. With a low potential difference between the electrodes the
luminosity would be expected to appear gradually, and with a large potential difference
to appear more abruptly. It has already been stated that this is what was observed.
The conditions which decide whether the luminosity appears round the anode or
the cathode need further investigation and will form the subject of future research.

(5) Summary of Results, and Conclusion.

The results contained in this paper may be summarised as follows :—

1. The experiments with a platinum cathode show that the negative leak from
platinum in helium or argon at low pressures is practically the same as in air or
oxygen. The variation of the negative leak per square centimetre with the
temperature can be expressed by an equation of the form used by H. A. WILSON and
by O. W. BICHABDSON, viz., x = A^e"*472*, where x is the current in amperes, 6 the
absolute temperature, and Q and A are constants. The values of these constants for
a cathode well cleaned with nitric acid in helium at a pressure of 3 '236 millims.
with a potential difference of 40 volts between the electrodes are Q = 1'22 x 10s,
A= 1-55 xlO9.

2. Attention has been drawn to a curious increase in the negative leak caused by
allowing the cathode to stand for some time with a very low gas pressure in the
apparatus. This increase seems to be caused by the appearance of a dark substance
on the surface of the platinum cathode. The substance is probably produced by the
action of the mercury vapour on the platinum. It could be driven away by long
continued heating of the cathode.

3. The negative leak from calcium is enormously greater than from platinum at
the, same temperature. As with platinum, the variation of the leak with the
temperature can be expressed by the equation x = A0*r"**, but the observed values
of the negative leaks from calcium at different temperatures do not so closely agree

NEGATIVE ELECTRICITY FROM HOT CALCIUM AND FROM LIME. 169

with the values calculated from the equation as is the case with platinum. This is
probably clue to the greater experimental difficulties attending the use of calcium.

4. On oxidising the calcium on the cathode to lime there is an enormous increase
in the negative leak, the leak from a lime cathode at 950° C. being about the same as
the leak from calcium at 140U°C. The variation of the negative leak from lime with
the temperature roughly obeys the Wilson-Richardson law, but the leak at any fixed
temperature is not constant, but gradually decreases with continued heating. This
diminution is not due to the same cause as the diminution of the leak from a new
platinum wire. In the case of platinum the decrease is caused by the gradual
evolution of gas — probably hydrogen — occluded in the metal. With lime it seems ta
be due either to a spluttering away of the lime from the surface of the platinum or to
a change in the nature of the lime itself. In this connection it is interesting to note
that a piece of lime subjected to a strong heat glows very brightly at first, but
gradually loses this property of glowing when kept continuously at a high tem-
perature. It is not improbable that there is some connection between these two
phenomena. Experiments are at present being made with this idea in view.

5. The following are the values of the constants Q and A for platinum, calcium,
and lime respectively, obtained from observations of the negative leaks under a
potential difference of 40 volts in helium at a pressure of 3"236 millims. :—

Cathode. Q (calories). A.

Platinum 1- 22xlO» l'6x!0«

Calcium 7-29X104 1-7 x 104

Lime 9-58x10* 6-4 xlO11

The constant Q represents the work done by a gramme molecular weight of
corpuscles in escaping from the surface of the cathode. We see from the numbers
given above that this is least in the case of calcium, but owing to the great variations
in the different values of Q obtained for lime and for calcium we cannot lay much
stress on the difference between the mean values for these two cathodes given above.
Tt would, of course, be expected that the corpuscles would escajn; more easily from
the metal than from the oxide, for we should expect the presence of the electro-
negative atom of oxygen in the molecule to act as an attracting force tending to
i-etaiu the escaping corpuscle.

6. The greatly increased leak obtained by oxidising the calcium cathode into lime
is due to an enormous increase in the value of the constant A. Reasons have been
given for thinking that this constant cannot be proportional to the number of free
corpuscles per cubic centimetre of the cathode as follows from RICHARDSON'S theory
of the negative leak.

7. The negative leak from lime in hydrogen is much greater than that in air or
helium.

8. When the current density through the discharge tube reaches a certain value
(only obtained with a calcium or line cathode) the gas becomes luminous. This

VOL. CCVII. — A. Z

170 DISCHARGE OF NEGATIVE ELECTRICITY.

luminosity appears abruptly if the temperature of the cathode is high and the
potential difference between the electrodes is gradually increased, or if a large
potential difference is used and the temperature of the cathode is gradually raised.
The abrupt appearance is accompanied by a large increase of the current passing
through the tube. The luminosity cannot be obtained with a potential difference of
less than 18 volts, which is probably the value of the anode fall of potential. With
small differences of potential (between 18 and 40 volts) the luminosity appears quite
gradually as the temperature of the cathode is raised, and without any sudden
increase in the current passing.

With low potential differences the luminosity appears sometimes round the anode
and sometimes round the cathode. In the former case it may be caused to leave the
anode and to appear round the cathode by increasing the potential difference. This
inversion of the appearance of the discharge is always accompanied by a large increase
in the current.

In conclusion, I wish to say that my best thanks are due to Professor J. J. THOMSON
for his advice and interest in these experiments, which were carried out at the
Cavendish Laboratory.

V. The Gravitational Stability of the Earth.

/>>/ A. E. H. LOVE, F.K.S., Sedleian Professor of Natural Philosophy in the

University of Oxford.

Received February 16, — Read March 14, 1907.

CONTENTS.

PART I.

Page

Introduction 171

Statement of the mathematical problem 174

Solution of the differential equations by means of spherical harmonics 178

Adjustment of the harmonics to satisfy the boundary conditions 188

The frequency equation and the condition of gravitational instability 191

Instability in respect of radial displacements 193

Instability in respect of displacements specified by harmonics of the first degree 197

Stability in respect of displacements specified by harmonics of the second and third degrees . . 202

Summary of the solution of the mathematical problem 210

Application to the problem of the gravitational stability of the earth (propagation of earth-
quake shocks) 213

PART II.

A past state of gravitational instability as a reason for the existing distribution of land and

water 217

Illustration of the nature of a hemispherical distribution of density 218

Effect of rotation upon a planet with such a distribution of density 221

Effect of certain external forces 225

The problem of the shape of the lithosphere 225

Spherical harmonic analysis of the distribution of land and water 226

The continental blocks and oceanic regions as expressed by means of spherical harmonica of the

first, second, and third degrees 236

Geological implications of the theory 238

PART I.
INTRODUCTION.

1. IF in a gravitating body there occurs a displacement which involves alteration of
density, there must be a tendency for the material to move towards the places where
the density is increased, and away from the places where the density is diminished.
The effect of this tendency, if it were not held in check, would be to accentuate local
VOL. ccvn, — A 417, Z 2 31.5.07

172 PROFESSOR A. E. H. LOVE ON THE

alterations of density. In any body the tendency is partially held in check by the
elasticity of the body, and, in particular, by the elastic resistance which the body
offers to compression. If this resistance is sufficiently great, the body is stable, iu
spite of the tendency to instability which arises from gravitation. It is important to
determine the conditions of stability for bodies of various forms and constitutions,
with various distributions of density. The problem of the stability of spherically
symmetrical configurations of a quantity of gravitating gas has been investigated by
J. H. JEANS,* and he has drawn from his investigations some interesting conclusions
in regard to the course of evolution of stellar and planetary systems. In a subse-
quent memoirt he proceeded to investigate a similar problem in regard to gravitating
bodies of a. more coherent character. A gravitating solid body, such as a planet may
be conceived to be, might exist in a spherical shape with a spherically symmetrical
distribution of density. In the absence of gravitation there could be no question of
instability. The effect of any local condensation would be to set up vibrations, and
the frequency of the vibration of any spherical harmonic type would depend upon the
elasticity of the material. If the resistance of the material to compression is suffi-
ciently high the stability persists in spite of gravitation. There are thus two
competing agencies : gravitation, tending to instability, and the elasticity of the
material, tending to stability. In a general way it is clear that, as the elasticity
diminishes, the frequency of vibration of any type also diminishes ; and, if the
frequency can vanish for sufficiently small elasticity, the planetary body possessing
such elasticity cannot continue to exist in the spherically symmetrical configuration.
The problem is to determine the conditions as regards elasticity in which the
instability occurs.

A grave difficulty presents itself at the outset. In the equilibrium configuration
the gravitating planet is in a state of stress ; and, in a body of such dimensions as the
Earth, this stress is so great that the total stress existing in the body when it vibrates
cannot be calculated by the ordinary methods of the theory of elasticity. In that
theory it is ordinarily assumed that the body under investigation is in a state so little
removed from one of zero stress that the strain, measured from this state as a zero ol
reckoning, is proportional to the stress existing at any instant. In order that this
assumption may be valid, it is necessary that the strain which is calculated by means
of it should be so small that its square may be neglected. Now if we apply the
equations of the ordinary theory to the problem of a solid sphere strained by its own
gravitation, and if we take the sphere to be of the same size and mass as the Earth,
and the material of which it is composed to possess moduluses of elasticity as great as
those of ordinary steel, we find that the strains may be as great as •£, and thus the
strains are much too great for the assumption to be valid. The initial stress existing

* " The Stability of a Spherical Nebula," London, 'Phil. Trans. Roy. Soc.,' A, vol. 199 (1902), p. 1.
t J. H. JEANS, " On the Vibrations and Stability of a Gravitating Planet," London, ' Phil. Trans. Roy.
Soc.,' A, vol. 201 (1903), p. 157. Quoted below as " JEANS (1903)."

GRAVITATIONAL STABILITY OF THE EARTH. 173

iu the gravitating planet, the stress by which the self-attraction of the body IK
equilibrated, is much too great to permit of the application of the ordinary theory.
The same difficulty presents itself in every problem concerning the elasticity of a
gravitating planet, for example, in the problem of tidal deformation or of the stress
produced in the interior by the weight of continents. In these problems the difficulty
was turned by Lord KELVIN* and Sir G. H. DARWIN! by taking the modulus of
compression to be much greater than that of any known material, in other words, by
taking the material to lie incompressible. Their object was to determine the degree
of rigidity which must l>e assigned to the Earth, and for that object it is permissible
to turn the difficulty in this way. When the problem is that of gravitational
instability this artifice cannot be adopted, because the whole question is that of the
degree of compressibility which is admissible if the gravitating planet is to be stable
in a spherically symmetrical configuration. The artifice adopted by JEANS (1903)
consisted in annulling the initial stress by introducing an imagined external field of
force to equilibrate the self-attraction of the planet.

The problem thus posed is an artificial one, which may, nevertheless, throw light on
the actual problem. When the initial configuration is taken to be one of uniform
density, the analysis of the problem is of the same kind as that which presents itselt
in the problem of the vibrations of an elastic sphere, a problem which has been worked
out very completely by H. LAMB.| The determination of the effect produced by
gravitation in lowering the frequencies of the various modes of vibration is reduced to
a question of troublesome analytical computation. JEANS worked out the problem on
the basis of the ordinary theory of elasticity, using the elastic constants X and p. of
I ..\ M f:. The constant /* is the niodulus of rigidity, and the constant X is such that
X+f/it is the modulus of compression. In the case of the Earth the values of these
constants can be inferred from the observed rates of propagation of the various types
of disturlwuice which are perceived as earthquake shocks. He -concluded that, when
the proper values are attributed to these constants, the Earth must be held to be in
a state far removed from one of gravitational instability ; but he suggested that, if
the resistance to compression was at one time considerably smaller than it is now, the
spherically symmetrical configuration would then have been unstable ; and he held
that there are traces of the instability in the distribution of land and water on the
surface of the globe.

The actual problem differs from this artificial problem in the mode of balancing of
the internal gravitation. Lord RAYLEIGH§ has proposed a method of meeting the
difficulty as to initial stress. He proposed to consider the stress in the vibrating

* See, in particular, KKF.VIN and TAIT'S ' Natural Philosophy,' Part II., S§ 833-846, Cambridge, 1883.
t " On the Stresses caused in the interior of the Earth by the weight of Continents and Mountains,"
London, 'Phil. Trans. Roy. Soc.,' 173, 1882, p. 187.

} "On the Vibrations of an Elastic Sphere," London, 'Proc. Math. Soc.,' 13, 1882, p. 189.

§ "On the Dilatational Stability of the Earth," London, 'Proc. Roy. Soc.,' A, 77, 1906, p. 486.

174 PROFESSOR A. E. H. LOVE ON THE

gravitating sphere as compounded of two stress-systems : a hydrostatic pressure by
which gravitation would be balanced if the sphere were in equilibrium, and an
additional stress. He proposed to measure the strain, not from the unattainable
state of zero stress, but from the equilibrium state ; and he proposed to take the
additional stress to be determined in terms of the strain by those equations which are
commonly used in the theory of elasticity. To simplify the problem he proposed to
take the material in the equilibrium state to be homogeneous and the elasticity to be
isotropic, so that the equations connecting the additional stress and the strain are of
the same form as the ordinary stress-strain relations of isotropic elasticity. In
justification of the proposed procedure he brought forward theoretical considerations
founded upon the general theory of energy, and other evidence drawn from an
interpretation of the experimental results in regard to the behaviour of elastic solid
bodies. It is not too much to say that all the evidence there is, is just as strong in
favour of Lord RAYLEIGH'S proposed method as it is in favour of HOOKE'S law, in the
sense in which that law is applied in the ordinary theory. The only objection which
can be raised against the method, an objection mentioned by Lord RAYLEIGH himself,
is that the body to be treated is certainly not homogeneous, and possibly not isotropic.
When the proposed method is adopted, the density and the moduluses of elasticity
must be taken to have their mean values. The justification for treating the values of
these quantities at any point as equal to the mean values, is that it is advisable in
the first instance to work out the simplest case.*

In the first part of this paper the mathematical problem proposed by Lord RAYLEIGH
is worked out ; and the conclusion is drawn that the effective moduluses of elasticity
of the Earth, in its present state, are sufficiently great for a homogeneous spherical
configuration to be thoroughly stable. The second part of the paper is devoted to
developing the consequences of supposing that the elasticity of the material of the
Earth was once muclrless than it is at present.

Statement of the Mathematical Problem.

2. We have before us a perfectly definite mathematical problem, which may be
stated as follows : — A sphere of radius a, and of uniform density p0, is in equilibrium
under its own gravitation, and the stress within it is hydrostatic pressure of amount
Po at a distance r from the centre. When any small disturbance takes place, so that

* It may be observed that the method advocated by Lord RAYLEIGH is the same, except for a slight
modification, as that which was used in the second edition of my " Treatise on the Mathematical Theory
of Elasticity," Cambridge, 1906, in the discussion of the statical problem of a gravitating sphere held
strained by external disturbing forces. The modification consists in the assumption, which was there
made, that the material might be treated as incompressible. If this assumption is not made, the analysis
becomes much more difficult. An earlier indication of the method will be found in a paper by J. LARMOR
Cambridge, 'Proc.. Phil. Soc.,' 9, 1898, p. 183.

GRAVITATIONAL STABILITY OF THE EARTH.

175

the particle which was initially at (x, y, z) is displaced to (z+ti, y+v, z+w), the stress
is specified by six stress-components X,, Yy, Z., Y,, Zz, X,, and these are connected
with the initial pressure /)„ and the displacement (u, v, w) by the formulae

3t*

ov

. . (1)

where X and /* are constants. It is required to form the equations of vibration, and
to solve them, so as to determine the character of the modes of vibration and the
equation for the frequencies, and, in particular, to ascertain the relations which must
hold among the quantities X, p., p^, a in order that any frequency may be reduced to
zero. We proceed to express this problem in terms of a system of differential
equations which hold at all points of the body, and a system of special conditions
which hold at all points of the undisturbed surface.

3. In the equilibrium state the potential V0 at any point is given by the equation

-r»), ......... (2)

where y is the constant of gravitation. The equation of equilibrium is

_!3£o+3V, ...... (3)

po dr 3r
or

|r=-^*V. .......... (4)

Since p* = 0 at the surface r = a, the value of p* at any point is given by the
equation

^lirypo'Ca'-r*) .......... (5)

When the sphere vibrates, the equations of motion are three equations of the type

3s?* 3V . 3X, _,_ 3X, ^ 3Z,
/>3rf = P -5- + -5-* + -*-*+ -3-*, ....... (6)

3r r dx ox ay cz

where p is the density, and V the potential, in the disturbed state. In the left-hand
members of these equations we may ignore the distinction between p and fv In the
right-hand members we may put

p^oO-A) ....... .-. . ... (7)

176 PROFESSOR A. E. H. LOVE ON. THE

where A is the dilatation expressed by the formula

.3_wch^3M>

= 3* ty ~Sz' ' ' ' (8

Further, we may put

V = V0+W '•, (9)

where W is the additional potential due to concentration of density at internal points,
and to displacement of mass across the initial bounding surface. We may neglect
terms of the type /30A3W/3a:. When we substitute for XZ)... from equations (1), and
make these simplifications, the equation (6) becomes

On omitting the terms which cancel each other in virtue of equation (3), we have the
first of the three equations (10) below. The remaining two of these equations
are obtained in the same way. Thus we have the equations of vibratory motion in
the forms

%•> "»>

In addition to these equations we have the equation connecting the potential with
the density in the form

V2W = 47ry/>0A ........... (11)

The system of equations (10) and (11) are the differential equations of the problem.*
4. Besides satisfying the differential equations (10) and (11), the additional
potential W and the components of displacement u, v, ^v must also satisfy certain
conditions at the surface r = a. Let U denote the radial component of displacement,

so that

Ur = xu+yv+zW) ... ...... (12)

and let U0 denote the value of U at r = a. The potential W is that due to a volume
distribution of density — p0A, together with that due to a superficial distribution /30Ua

* In the problem as formulated by JEANS, when the self-attraction of the body is balanced by an
external field of force, the equations of vibratory motion differ from those which are obtained here by the
omission of the terms such as ^iryp^xA. In Lord KAYLEIGU'S paper already cited, the equations given by
JEANS are discussed in accordance with the analysis which was developed by LAMB in the paper on the
vibrations of a sphere.

GRAVITATIONAL STABILITY OP THE EAKTH. 177

on the surface r = a. By the method of spherical harmonics we can, when W is
known, write down the expression for the function W(u) which is the potential at
external points of the same distribution. The surface characteristic equation gives

This is one of the conditions which must be satisfied at the surface r = a. To
obtain the other conditions which must be satisfied at this surface, we observe that
the disturbed surface r = a + U« is free from traction. If /, m, n denote the direction
cosines of the outward drawn normal to this surface we have three equations of the

type

, = 0,

which hold at the surface r = a + Ua. If in this equation we substitute for XT, ...
from equations (l), we see that in the terms containing u, ... we may replace I, ... by
the approximate values x/r, yfr, z/r. The only term which does not contain u,... is
the term — Ip0 arising from —IX^ Now p0 vanishes at r = », and therefore at
r = a + Ua we have

to the first order in u, v, w. Hence in this term also we may replace / by x/r. On
substituting for^>rt from (5) we find that the equation

v , cu

must hold at the surface r = a. By an easy transformation this equation becomes
the first of the three equations written in (14) below. The remaining two of these
equations are obtained in the same way. The equations which must hold at the
surface r = a are therefore equation (13) and the equations

J_(Ur) +,- |H -u +| 3!fiLxUr = 0,
ox or p

JL(Ur)+r — -r+|E3ieL«Ur = 0, }- (14)

oy < r p

A C/ /TT \ Utv A TfrfJt\ TT

-?A + =-(Un+r;r tp+J ir° z\Jr = 0.

ft 01 O1' fJ.

These e<juations can be interpreted in the statement that the traction on the mean
sphere is a pressure equal to the weight per unit of area of the material heaped up to
form the inequality Ua.

VOL. ocvn. — A. 2 A

178 PROFESSOR A. E. H. LOVE ON THE

5. We shall now suppose that the system executes a normal, or principal, vibration
of frequency p('2ir, or, in other words, that every component of displacement is
proportional to the same simple harmonic function of pt. The equations of vibratory
motion become three equations of the type

0, .... (15)

where W satisfies the equation (11). The solutions of this system of equations (11)
and (15) must be adjusted to satisfy the conditions (13) and (14) at r = a. These
conditions can be satisfied only if p has one or other of a certain infinite set of values,
which are the roots of the frequency equation. The problem of gravitational
instability is solved when we find the conditions that one of the values of p may be
zero.

Solution of the Differential Equations by Means of Sphencal Harmonics.
6. We introduce the notation

The equations of motion (15) become three equations of the type

x
and the equation v2W = 47ry/30A becomes

V3E = A; (18)

in these equations A stands for

dx dy dz '

By differentiating the left-hand members of the equations of type (17) with respect
to x, y, z, respectively, adding the results, and simplifying by means of (18), we
obtain the equation

=0 ......... (19)

The method of solution of the problem is this : — We seek first a solution of the
equation (19) in which A has the form

, .......... (20)

where tan is a spherical solid harmonic of positive integral degree n, and /„ is a

GRAVITATIONAL STABILITY OF Till. KAHTII.

179

function ol r which is such that i*fH is finite at all points within r = a, including the
origin r = 0. We seek next to determine E in the form

(21)

where E,(r) is a certain function of r which is such that »•"£„ is finite at all points
within ?• = a, including the origin r = 0, and F, is a spherical solid harmonic of
degree n. The equations of motion of type (15) then become three equations of the
type

in which we must have

= 0,

3it

. . (22)
• (23)

It appears on trial that we can obtain a solution in which

U = «, + Mj + «3, V = f, + t',+ t'3, W =

where u,, r,, w, satisfy the system of equations

. . (24)

*. . . . (25)

also tig, v,, wa satisfy the system of equations

, , s _ .

« — ' o — ' "5 " »

3x dy 3z
and Us, r,, tw3 are a complementary solution of the system of equations

3x 3y 3z

7. The sets of functions uu vlt wl and u3, vt,

2 A 2

(26)

. . . . (27)

can be any particular solutions

180 PROFESSOR A. E. H. LOVE ON THE

of the systems of equations (25) and (26). It appears on trial that «,, vlt w^ can
have the forms

(28)

where PM(r) and Qa(r) are certain functions of r. Also it is clear that ua, va, wa can
have the forms

n\ 3s2v8FM /oftX

*— *- • (29)

Further, the forms of u3, v3, w3 are known from the analysis of the problem of the
vibrating sphere which is free from gravitation. We have

I, • (30)

\ G 5 Qtl / CtJu ii T"~ .L O*t/ \i / _

where

T»\ ' \ 1.,. Jl.,.l \ /.„ /' \ *

XB and (f)n are spherical solid harmonics of degree n, and the expressions for va, u'3 are
to be obtained from the expression for u3 by cyclical interchanges of the letters x, y, z.
It appears that to a single term fno>n in the expression for A there corresponds a
definite term Fn in the expression for E. Further, when we form the boundary
conditions, it appears that the terms of ua, va, wa which contain •%„ represent a free
vibration, and the frequency of this vibration is determined by the same equation as
if the sphere were free from gravitation. Also it appears that to any term fn<an in
the expression for A there corresponds a definite function <£„ in the expression for
Us, v3, ws. The solution expressed by a single term /„&„ of A and the corresponding
terms of (MI, v,, w^, (ua, va, wa) and (u3, t*3, «03) determines a normal mode of vibration.
We shall therefore omit x>., and reduce all the summations to typical terms.
8. If in the equation

(Va+ h") A + G«3A + «V?- = 0 . ... (19 bis)

or

we put/B(r) . to, for A, we find that^,(r) must satisfy the equation

=

dr r dr \ dr
or

}/n = 0 ..... (32)

GRAVITATIONAL STABILITY OF THE EARTH. 181

This equation is a linear differential equation of the second order ; and the forms of
the coefficients show that the point r = 0 is a critical point, and that there is no
other critical point at any finite value of r. If we seek a solution in series of

the form

/. =

we find the " indicial equation "

m(m-l)+2(n+l)m = 0,

from which either in, = 0, or m = — (2n+l). We must take the series for which
m = 0, because ?•"/» must be finite at r = 0. Further, the form of the equation shows
that this series contains even powers of r only. We assume, therefore, for fn

the form

/„ = A [H-a,r*+a4r4+... + afcr*1 +...],

where A is an arbitrary constant, and then we find the sequence equation

= 0,

or

h'

°fc+' = a"(2K+2) (2K+2n+~3)
Hence we have

f - . A fl - *' t^J-AKr*. itf±i!L+6) Q_{ A'_±(n±8) *} 4_
/' L 2.(2n+3) 2.4(2n+3)(2n + 5)

. y{A'+(n+6)<'}{y+(n + 8)a'}...{A'+(n + 2* + 4)Jia} ,. 1
2.4...2»c(2n-l-3)(2n-f5)...(2n + 2K+l) "J"

The series is convergent and represents the function fn for all finite values of r.
9. We must next determine the function E. (r) from the equation

or

, ( .

"

r dr
or

We introduce an intermediate function 0n (r) by the equation

Then

6. = \rfndr

. ......... (35)

= C + A 4r»--" ^r* +

2.4.(2n+3) 2.4.6(2n-f3)(2n-»-5)

(n-HS)^}^^
)(2n-»-5) "J '

182 PROFESSOR A. E. H. LOVE ON THE

where C is constant of integration. Then we have

and therefore

, Ar •* {A2-f(n+6)sy

l 1.2.(2n + 3) 2 . 4 (2n + 3) (2ft + 5)

where 0' is a constant of integration. Since »-"En is finite at r = 0, the constant C'
must be zero ; but the constant C is in our power, and we may choose it in any way
that is convenient. The term contributed to E by C is (2n+l)~1 C<uB, which satisfies
LAPLACE'S equation, and therefore any change in the chosen value of C is equivalent
to borrowing a term of Fn to make up a term of £„<»„.
Now the series

_ + ._

2.(2n+l) 2.4(2« + l)(2n+3)

satisfies the equation

and therefore, if we take for C the value

P - 2n+l .

fca+(w+4)s2

the function 0n satisfies the equation

We shall choose this value for C, and thus we shall have
2n+1

. i^ 4

2.4(2n+3)

.Af ___ 1_ ^ {tf+fri+G)*}!*

L ^+^+4)^ 2(2n + 3) 2.4.(2n + 3)(2n+5)

(2n + 3) 2.4.(2n + 3)(2n+5)

/ v

/ \,+i \i r \n f \rt, ofs |...^< i- {/t,-r6K-r6j a j Q, /^T^

2.4...2/c(2n+3)(2n-l-5)...(2n + 2ff-l)
and

2. 4...2K(2n + 3)(2n+5)...|
The function En satisfies the equation

I dr

GRAVITATIONAL STABILITY OF THE EARTH. 183

It will be convenient presently to have observed that the equation derived from
this one by differentiating the left-hand member with respect to r can be written

10. Tljp forms of «„ v,, wt and u^, va, w3 have been put down and it remains to
determine ut, vlt «>,. We have a system of three equations of the type

We express x<aH in the form

(42)

and then the above equation becomes

2n+ 1 8* y LW / »• rfr A" r

We seek solutions of the system of three equations of this type in the forms of
the type

in which P, and Q, are functions of r. We find

Hence the assumed forms can be adjusted to satisfy the equations if P, and Q.
satisfy the linear equations

2ndP
"

184 . . PROFESSOR A. F, H. LOVE ON THE

The right-hand members of these equations can be simplified by introducing the
functions #„ and r~l dEJdr. We have

1 d6n r d /Id

2n+l dr ~ r dr 2n+l dr \r dr

1 (#0* , 2n dflA
~2n-t-lVdr2 r dr/'
and

d6,
r-r-!

2» +

and thus the equation for PB becomes
d'P 2ndP pp

T > r A- J: „

. , ,3

""

2n+l\dr2 r dr/ 2»+lA2\dra r

I

2n+
by the equation (36). Hence we have

where P'H satisfies the equation

r dr
Again, we have

1 dfn = 1 d Jd*E. 2(n+
r cZ?1 ?• dr 1 d?-8 r

+

r dr / J r dr J

r dr V?1 dr / '
and

,3 dE. _ d'E. 2(n + l) dEB 3 dEB

•/• • ~T 1~5 1 7 ' J

r «r ar* ?• dr r dr

_rd .

dr \r c//% / ?• dr

GRAVITATIONAL STABILITY OF THE EARTH. - 185

%

and thus the equation for Q. becomes

r d

^L (i fJ^\ + 2 (n ±2) _<L (I ^L- VI

__ _

2n+l i<h*\i' dr r dr\r dr

1 V([d*_ A rfEjA , 2_(n+ 2) ^ /I </E.\ . _rf /I dE.

2n+ I A'.U^ \r 3r"/ / rfr \r rfr / dr \r dr

1 [fP /I rfE.\ + 2(n+2) rf /I «ffi.\ +p /I rfjE.
2n+ 1 [_d>* \r dr I r dr -r dr / ' \r dr

by the equation (40). Hence we have

where Q'. satisfies the equation

11. To find the form of P'H, we assume

^A(^+^+^+...) ....... (51)

Then equations (48) and (37) give

2.4...2K(2n+3)(2n+5)...(2»+2»t-l)

VOL. CCVII. A. 2 B

186 PROFESSOR A. E. H. LOVE ON THE

*

By these equations p0 is left arbitrary, and p3, p4... are determined when p0 is
chosen. As we need a particular integral only of the equation for P'n, we may choose
p0 in any way that may be convenient. We shall put p0 = 0. Then

,

a °' 2.4(2n + 3)

p» =

(52)

To find the form of Q'H, we assume

4+-) (53)

Then equations (50) and (38) give

As before, ^0 can be chosen arbitrarily, and then qa, qt>... are known. We observe
that if we put

9o = ' ft*-' '•" f/a'= -"" ^

the sequence equations for the q's are transformed into the sequence equations for the
p's, beginning with the equation containing pa, pt. We shall therefore choose q0 to
be (2n + 3)~12p3, and then

->-4+...j . (54)

2n+5 2n+7

GRAVITATIONAL STABILITY OF THE EARTH. 187

This choice of the q's amounts to subjecting the functions F. and Q'. to the
equation

^rf ........ (55)

dr

To see that this equation is compatible with the differential equations (48) and (50)
for P', and Q',, we observe that

_
dr ~ r**+ldr \r dr

and that from the equation (48) we can form the equation

d „ 2(n + l)1 /- .rfF,\_. n+1 P,d [ ..+,/! dE.\1
» J\ dr]-2n+lh> dr\ \rdr)\'

r dr
while from (50) we can form the equation

d" 2(n+l)d „ 2(n+l)1 f <* /^ip/i] . « *» . d/,/1 dE.
P r dr r> )\dr( ~ 2^1 h^ dr

12. We have still to satisfy the condition

If we form the expression in the left-hand member from the expressions of the
type (44) for «,, vlt «?,, we have

a*, at, ^ = jn dp. _ IHO [rf

3x 3y 3z lr rfr r*"^Jrfr

By means of the formulae (47) and (49) for P, and Q,, the coefficient of eu, in the
right-hand member of this equation is transformed into

1 f« ^ , n+l d_ / >,„ dE.\-\ n+1 f n
2n+ 1 [r dr r"+8 rfr \ </r /J r*"+' L»+ 1

The first term is equal to

[ [»/. + («+ 1)/.] or /.

and the second term vanishes identically in virtue of equation (55). It follows that,
with our choice of pt and qa, the equation (56) is satisfied identically.

13. We have now completed the determination of the forms of u, v, w in terms of
the spherical solid harmonics &»„ F,, <f»n, and of certain functions of r, viz. : f., On, E,,

2 B 2

188 PROFESSOR A. E. H. LOVE ON THE

P'». Q'». 'A*- Various relations between these functions have been noted incidentally.
It will be convenient hereafter to have noted the following properties of »/»,, (kr) : —

1
1.3.5. ..(2/1+1) I 2(2n + 3) 2.4 (2n + 3)(2n+5)

^ j _ ; .^ ;

A-r «(#?•) \

kr)}. . . . (59)

Adjustment of the Harmonics to Satisfy the Boundary Conditions.

14. In order to express Fn in terms of &>„ and <f>n we use the condition that
30 (EncoB + Fn) is the potential at points within the sphere r = a of a distribution
of volume density within the sphere and of surface density on the sphere. The
corresponding external potential is

where EB (a) is the value of EB at r = a. The surface density is /50Ua, where Ua is
the value at r = a of the radial displacement U. Hence we have the equation

which holds at the surface r = a ; it gives

Now

U = Pn-

2n-

.pl =U,
Jr = 0

It follows that the equation

Q

holds at the surface »• = a. Since this equation connects the values at r = a of three
spherical solid harmonics of the same degree n, it holds for all values of r, and gives

GRAVITATIONAL STABILITY OF THE EARTH. 189

a generally valid expression of F, in terms of CD. and <£„. By means of equations (35)
and (59) the equation becomes

15. The three remaining conditions which hold at the surface r = a are expressed
by equations of the type

Every term of the left-hand member can be expressed in terms of the spherical

solid harmonics

Sw. »,+, 3 / a). \ 9<ft, *,+3 3

aF' 3*1^7' ax' ai

We have

Also
rU =

+ 3^2^T)P[{^~"P'+(" + l)alQ"}' = a-^
and therefore

£<rU) = [nP.-

.(*r)J3*. ,+i 3 / *i M

— M-^5-' :r- (~ ^7 r

rfr ISx axVr***1/]

190 PROFESSOR A. E. H. LOVE ON THE

Further,

16. The equations which hold at the surface r = a can he arranged in such
forms as

A ^?+BraB+3— f^s-\+C ^+D r2"*3 — (-&±}= 0 (62)

" n *+ "H ** j

in which AB, BB, C,, DB are certain functions of a, viz.

c. = -.

^- •••.-• ••.,•.•.;:,• • • • (C5)

D" - da*-^- 4Ti'{affi*"'<k<) +'"A-«<te)}

» , /7 . ,,-v

to>" ' ' ............. (66)

In these formulae a is supposed to he written instead of r in the expressions for

/., ft,, P., Q..

GRAVITATIONAL STABILITY OF THE EARTH. 191

17. We may express A. and B, in more convenient forms by the use of the

identities

nP.-(n+l)aU = 0.-

M*-a
da ' aj

_, - -

a da

a da
We find

f 1
B"= ~i

'aJQ',,}. . . . (68)

We may also express C. and D,, in simpler forms by the use of the equations
connecting the t/» functions. We find

<70)

The Frequency Equation and the Condition of Gravitational Instability.

18. Exactly as in the problem of the vibrating sphere which is free from gravitation,
it follows from the equations of type (62) that we must have at once

A.«.+ 0.^ = 0, and B.w.+D^,, = 0 ...... (71)

and the frequency equation is of the form

A.D.-BLC. - o .......... (72)

The forms of all the functions which enter into the expressions of A,, B., C,, D. have
been determined.

To investigate gravitational instability, we have to determine the conditions which

192 PROFESSOR A. E. H. LOVE ON THE

must hold in order that the frequency equation may be satisfied by p2 — 0. When p*
vanishes, h2 and &2 also vanish, but the quantity tfjh2, which is (X+2/i)//ii, has
a determinate value. We may not, however, obtain the result which we seek by
first replacing P//t3, wherever it occurs, by (X + 2//, )///., and then putting h? and A.-2
equal to zero wherever they occur, otherwise than in the ratio Ifjh3. This pre-
cautionary statement is necessary because it appears from the formulae (69) and (70)
of § 17 that C,, and DB both vanish if h' and P vanish. Thus we ought to regard
the equations (71) as being equivalent to the equations

in other words, we ought to remove a factor F from the equation ABDB— BBCB = 0
before putting h2 = 0 and k? = 0. An exceptional case occurs when n = 1. In this
case CByt~2 vanishes when h2 vanishes, and it will appear that AB also vanishes with h2,
and the equations (71) ought to be regarded as equivalent to

= 0, B^ + D^-2 (jfcty) = 0,

and we must remove a factor hfk2 from the equation AiDj — B^ = 0 before putting
It2 = 0 and k2 = 0. . When we proceed in this way, the equation ABDB— BBCB = 0,
with the appropriate factors removed, and with h2 and k2 put equal to zero after
their removal, becomes an equation to determine «2a2, or |7ry/302a2/(X + 2/i). If the
equation has a real root', the value so determined for s2a2 gives a value of X+2/z. for
which instability can occur. Since P<f>n is a finite multiple of <an when X+2/n has any
such value, it is certain that the homogeneous spherical configuration really is unstable
for such values of X+2/A. If the value of X+2/4 belonging to the body is but little
greater than the critical value, the equilibrium is practically unstable ; for a large
displacement takes place if the sphere begins to vibrate according to the type
specified by the degree n of the corresponding spherical harmonic function. For
practical stability it is necessary that the value of X+2/u, should be well above any
critical value. The equation which yields the critical values contains the constant
(X+2/u.)//Lt as well as sV. It will be convenient to write

v =

x+^'jf ;-'• ; ' • ; * • • (73)

The value of v cannot be negative, nor can it be greater than f . If the POISSON'S
ratio (X/2(X+/it)} of the material is positive, v cannot exceed \. If the modulus of
rigidity \L were very small in comparison with the modulus of compression X+§/u,
v would be very small. If the velocity of propagation of waves of dilatation were
twice that of waves of distortion, v would be ^. This appears to be the most
appropriate value to assume in the case of the Earth (see § 40 below). Since it is
improbable that the ratio of the rigidity to the modulus of compression of the Earth
has diminished since the date of consolidation, it will be sufficient for our purpose to

GRAVITATIONAL STABILITY OF THE EARTH. 193

examine the two coses in which v = 0 and v = |. We have now to discuss the
conditions of gravitational instability in respect of the values 0, 1, 2, ... of the number
n which specifies the type of vibration.

Instability in Reaped of Radial Displacements.

19. The case in which n = 0 is the case of a sphere vibrating radially. This case
is not very easily included in the foregoing analysis, and it is very easy to investigate
it independently. Let U denote the radial displacement. Then U is a function of r,
and we have

XTT VTT «TT 2

u = - U , v = 2. U, w = - U, A = -- + -- .

r r r fir r

2U

, = --
fir

We go back to the equations (15) of the type

where W, the additional potential, is a function of r. This equation is

d /rfU . 2U\ . \ld* 2 eZ\/U\ Ox d/U
r _- + _ )+n\x -7-5 + - -^- -1 + 2- -3- (—
r\dr r I [ \dr r dr/ \r ] rdr\r

Now

2 dW

.,0. . (74)

r r dr

fd\J A 2U\

, , --- j— = 47J-yp0( -= — h- ),
or r dr \dr r J

and therefore we may write

dW
•^.-iryipfcU+B,

where

j- + - R = 0, or Rr* = const.
r r

Since dW/dr is finite at r = 0, we must have R = 0, and thus equation (74) becomes
after division by (X + 2/i) a;/r

2+ +w+3s>U + h>U = 0,. . . . (75)

r dr r* \ dr /

where .f* = j7rypoS/(X + 2/x) and h* = papt/(\ + 2p.). This equation can be solved by
means of a series, which is convergent for all finite values of r, in the form

U - A r -

235 2.4.3.5.7

I i ( .

VOL. CCVII. — A. 2 C

194 PROFESSOR A. E. H. LOVE ON THE

where A is an arbitrary constant. The second solution of the differential equation
for U becomes infinite at r = 0, and thus the above is the most general form for U.

20. The condition that the surface ?• = a + \Ja is free from traction can easily be
shown to be the condition that

= 0
r

at r = a. Hence we have

(77)

where v = /u/(A.+2/i), so that 2A/(\+2/*) = 2 — 4i/. The frequency equation is
therefore

The condition of gravitational instability is obtained by putting A2=0. It is

)... = o. (79)

21. The coefficient of s^a2" in the left-hand member of (79) is

or

(-)'ff 1 , 1 _ 1

2 Ll3.5...2K-l 3.5...(2»c+l)j

2 l3.5...(2K-l)

v -- • -- --

(2»f-l) 3.5...(2ic + l) 3.5...(2K+3)JJ'

and the equation (79) can be written

where x is written for sa. Now we have

(I I ~3 5

oe'^ = e.-(,-| +^

and therefore

1 _ ^ + ^!_ _ . . . = x-*e-v ('
33.5 Jo

GRAVITATIONAL STABILITY OF THE EARTH. 195

and the equation may be written

-5)-(I-|-))]^"f.>& = °-- • • <80>

22. The left-hand member of equation (80), being equal to

is positive when x = 0. To determine its sign for large values of x we observe that

and therefore there is an asymptotic expansion* for xe~|x* I e**'dx when x is large in
the form

. 5ar"+3 . 5 . 7x~8+ ... .

Hence the expression in the left-hand member of (80) is asymptotically equal to

The term of highest degree independent of v is — 2x~* ; the term of highest degree
containing v is SKC'*. It follows that the expression is always negative when a; is
sufficiently great. The expression therefore changes sign for some positive value of x,
and the equation (80) has at least one positive root.

23. When v = 0 the equation (80) becomes

If x* < 1 the left-hand member is necessarily positive. We shall take x* > 1 and
write the equation

XT— 1

Let y denote the left-hand member of this equation. Then we have

dx

Since this expression cannot vanish, the equation cannot have more than one positive
root.

* For the suggestion that this stop might prove useful in demonstrating the existence of a root of
equation (80) I am indebted to Mr. G, H. HARPY, Fellow of Trinity College, Cambridge.

2 c 2

196 PROFESSOR A. E. H. LOVE ON THE

Again, when v = %, the equation can be written

3+-W3- V^l^-z^r^**^} = 0,
or \ ar or/ I Jo J

or

where the left-hand member is certainly positive when a^ < 1 ; also the differential
coefficient of the left-hand member with respect to x is

and this expression cannot vanish for any value of x which is greater than unity.
Hence the equation (80) cannot have more than one positive root.
24. Now take v = 0, and write the equation (80)

When y? = 4, we have

-r2 -r4

2

3 3.5 V 33.5

44123 47

3. 5. ..13 3.5...15V 177 3. 5. ..19
and

1 _ 1 _ 45045
a?-l" 3 3. 5. ..13'

-I)-...,

Hence, when y? = 4, the sign of the left-hand member of the equation (81) is plus.
When X* = 5, we have

. ..1-1+ >

3 3.5 3 3.5

20056 { . 15625 L 5

Also

3.7.9.11.13.3 3.7.9.11.13.3.17V 19

58 / 5 \

f 3. 7. 9. 11. 13. 3. 17. 19.21 \ ~23/ + ""

15625 Ii> 677,

17 19
and therefore

5 . 52 20733

3 3.5 3. 7. ..13. 3'

but

i(3.7...l3.3) = 20270 + £ < 20733.

Hence, when x2 = 5, the sign of the left-hand member of equation (81) is minus.
It follows that the value of x?, or tfo?, which satisfies the equation is between 4 and 5.

GRAVITATIONAL STABILITY OF THE EARTH. 197

25. Again, when v = \, equation (80) can be written

_8?JLL_ ./!_*+_£_. Uo (82)

3z*-23*+l I 3+3.5

If we put 3? = 4, the left-hand member becomes

41 \ 3 3.5
Now

_ia_l!_ 645461 4' /' 4 \

3 3.5~ =3.5...15 8.5...17V 19/

and

Hence 1 — -+ 7 — —...>--, and the sign of the left-hand member of (82) is
3 3.5 41

minus when x2 = 4. When we put a? — 3, the left-hand member of (82) becomes

__^1._ _ + __.
but

-3 + JL. = 3__?_ _?_ _9__/!_A. \
3 3.5 5 5.75.7 5.7. 11 V 13 /

7 5.7.11\ 13 •'/'
and

or the sign of the left-hand member of (82) is plus when 3? = 3. It follows that the
root of the equation (82) for x3 lies between 3 and 4.

Instability in respect of Displacements specified by Harmonics of the First Degree.

26. When n > 0, we have to calculate expressions for A,, B,, C,, D, from the
formulae of § 17. If n = 1, we have

JQ'1). • • (83)

Now if we put h* = 0 and IJ = 0, we find

f_ AT 7«V 9«V . v.^je+5)jj*-ofc 1

L 275+2T4T5~ ' 2.4...2K.5-J'

44

.s- /_ W

;

198 PROFESSOR A. R H. LOVE ON THE

and therefore, if we put h3 = 0 and P = 0 in A,, we get

A _a22AT s2a2 *V ,_v

~fr~5~[ 2 + 27I~ '

_

~2~ ' 2.4...(2K-2)'"

= 0.

It follows that A, vanishes to the first order in h2 and P, and therefore, as has t)een
explained, we must evaluate the limit of At/i~2 when /i2 and P vanish. We have to
expand the terms ofyi, s'6l and adP\/da correctly as far as A8; in calculating the
remaining terms of A1( we may put h2 and P equal to zero in E^ P\ and QV

The terms of /*, which are of the first order in h2 are

2r a2 7.90V /I 1\ . Y 7.9...(2*+S)j»-V /I 1 1 \ 1

L 2.5 2.4.5.7X7 9/ <V ' 2 . 4...2K . 5 . 7...(2*+3)\7 9 2K+5/'"J'

.(2*+3)
The terms of s2^ which are of the first order in h2 are

(-V+1 - + H

"

--- -

55s2 2.4.5 2.4...2/C.5.7." (2«+l)\7 9 ftc+8

Hence the terms of (fi + s20l) a,2/3v which are of first order in h2 are

2.4...2K

1 \1
+8/J-

Again, when we keep those terms only which are of the first order in h2 and P, we
find from (52) of § 11,

1 h2

__ __
2 . 4 . 5 5s2'

GRAVITATIONAL STABILITY OF THE EARTH. 199

and therefore the terras of adP'i/da which are of the first order in h* and If are

P

OV

where p,, ...have the above values, that is to say, these terms are

,

4a* /I .1,

r

2.4...(2*-2)7 9

,_
-

_ _

151/1.2.5 2.4.7 -l ' 2.4...(2K-2)(2ic+l)

It follows that the terms of the first order in h* and £* in

"

are

A/t

/tVfl a8 j^a4 /_\.J^L
15v L«* 2 2.4~ ^ 2.4...2K"*J

2AWfl^_lsV / x.^i 1 *"•'«" 1
* 15^ L5 2 72.4 2»c+32.4...2»c"'J'

or they are

2APaa/lgaa8_l s4a4 \
15*V \5 2 ~72.4

AJfcV -

15«»

Again, when we put /t2 and F equal to zero, we find
2A

E._2 _sa / y.+i__

El-5?L~ "2~ 2T14 2.4...2K

p/ 2A^ra* ji^ y'a* /_v.+ij^1f^_

15»U? 8.4 2.4.6 'V ; 2.4...2ie*"J>

, 2A*4 f a- 1 a4 1 sW / v ^"^

1= T5^L~5? 7 "2 92.4"'1 ; (2K+3)2.4...(2/c-2)"

and therefore the terms of the first order in h3 and If in

are

1., ,P 2A _|A, jZA f«V sV /_y+i_ **!?*-

~3 L 5?g 5^12.5 2.4.7 -V ; 2. 4...2*(2K+3)'" J J'

Hence the terms of A, which are of the first order in A* and A.* are

lAPa^-j^

and

A,_ Ao'e-i^

200 PROFESSOR A. E. H. LOVE ON THE

27. Again, when we put ha = 0 and F = 0, we find that

*h "•-*'•£- (ji+rffi) + - Tp +« -IT- - Q i ,
3i/u a da Ja

where

1 2 A f s2a3 s4a4 s2"*!*'
(yi + s20j) = 1 - + - ,.( — }*-'.

a"5a" 51 ~2~ f 2?4"'" ^~ ' 2. 4...2it"j '

1 __ _ .

da 15,/5 2.7 2.4.9 2. 4...2»e. (2«+5)

2 \7 / 2 \9

( ). f* ( 6 _!\ 1
' 2.4...2K\2/c+5 /" J

_
"

15i/ 5i/\5 7 2 92.4

Hence we have, when h2 = 0 and F = 0,

5

Also when n = 1 we have

Q, = GI =
and therefore

and we have also

and therefore

=

9 A r»°

^r(H"5 ^c-*"^ . . (85)

5v Ju

GRAVITATIONAL STABILITY OF THE EARTH. 201

28. In the case where n = 1 the condition of gravitational instability, viz.,

-1Jfc'il) = 0, becomes

_±Att ^ cf I A/ 2_ 1\ ,*, 2A, ^-.r^ e-k*dx\ = o (88)

105«V 8?15\ vl 5i>v Jo

But we have

and therefore the condition of gravitational instability becomes

3(me-^dx-e-^\38a+(^aY + (^--v}(m)&} = 0.
Jo I \2l) /

If now we put

= x9 = 2z',
the equation becomes

7rJ°

=0, . . . (89)

where the factor 2ir~* has been inserted because the expression in the square brackets
is tabulated in many easily accessible books.

Let y denote the left-hand member of the equation (89). When z is small, y is
small of order z&. In fact, we have

and when z is small, the first approximation to y is — (3— 4i>)z&. When z is great,

I e'^dt is approximately equal to £v/ir ; and thus y is positive when 2 is great. The
Jo

equation (89) has a root zero and at least one positive root. The zero root is
irrelevant to our problem ; it is introduced in transforming equation (88) into
equation (89). Now we have

^ = -zV 1(15-20,,) -|(19-20i/)zfl ;

and, since 15 — 20^ and 19— 20^ are positive when v<f, the expression last written
vanishes for one positive value of z. Hence it follows that the equation (89) has only
one positive root, and there is one and only one positive value of **a* which satisfies
equation (88).

By means of the tables it can be shown that, when v = 0, the root z lies between
1*9 and 2, so that sV lies between 7 '22 and 8. When v = £, the root z lies between
1-8 and 1'9, so that sV lies between 6'48 and 7'22.

VOL. ccvir. — A. 2 D

202 PROFESSOR A. E. H. LOVE ON THE

Stability in Respect of Displacements Specified by Harmonics of the. Second and

Third Degrees.

29. When n = 2 and A2 = 0, tf = 0, we have

. 2

02+3E2+3a2Q'2>
5

wnere

-.A J_ _ / y

"

_ _

6? 2.7 2.4.7.9 " ' 2.4...2/c.7.9...(2K + 5)'"'

-— a* Sy'a6 , y 8.10...('2>c+2)8ii'-4aiu 1

12s2 2.4.7 2.4.6.7.9 ' 2. 4...2/c.7.9...(2/c+3) "'J'

-22'

L_ «' ___ 8sV , y+1 8.10...(2<c+4)s2'-2a

___

'

5i> L 6.7.S2 2.7.9 2.4.7.9.11 >v ; 2 .4. ..2*. 7. 9. ..(2*4-7) "'J'
From these we find

A _ 2Aa2 [1 sV 8sV , y (2K+2)(2K4•4)siVl•

5>/L6 2.7 2.4.7.9 "^ ' 2 .4.6 .7. 9. ..(2*+5) "J

~~fr~ [677 ~ 2. 7. 9 + 2. 4. 7. 9.11~"^' ' 2 .4.6 .7.9 ...(2x4-7) "J

"~s3"L6~277 + 2.4.7.9~'"^~' 2.4.6.7.9...(2/c+5)'"J

-HAT-- — 4 8g4<t4 , y (2K4-2)(2<c+4)g2'a2l[ 1

s* [6 2.5 2.4.5.7 "^ ' 2 . 4 .6 . 5 .7...(2/c+3) "'J'

By means of the identities

^ 2_ 3

2/c + 3 (2/c + 3)(2/c+5)'

(2/c+2)(2/c+4) _ 1 6_ 15

(2/c+5)(2/c+7) " 2* + 5 (2/c+5) (2»c+7)'

(2/c+2)(2K+4) = _2
(2K+l)(2/c+3)

GRAVITATIONAL STABILITY OF THE EARTH. 203

we transform the series

1T1 _&_ 8x4 , y (2*+2)(2*+4).t* "I

5[_G 2.7 2.4.7.9 ' 2.4 .6.7. 9...(2K+5) '"J'

IT 1 x» 8s4 , _y (2* + 2) (2* + 4) a* "1

s|_6.7 2.7.9 2. 4.7.9. 11~ ' 2 .4.6 .7.9...(2»c+7) " J

[1 _^_ Sx4 x v. (2K+2)(2>c+4)a* "I

L6 2.5 2.4.5. 7~ "* ' 2 . 4 .6. 5.7...(2/c + 3) "J"

respectively, into the forms

Ll7l-24 J_\ aJ/i !_ 3 \ «/ 1 -2 S_ ]

ieLV ~3 3.5/~ ' \3 3.5 3.5.77 \3. 5 3.5.7 3.5.7. 9/ J

L[(l- A. J^-\-^(^ 6 15 \

16L\3 3.5 3.5.77 \3.5 3.5.7 3.5.7.'J/

J 1_ 6 15 \ ]

;\3.5.7 3.5.7.9 3. 5. 7. 9. 117 "J"

or

L[(i-*+.*L >\_2»1 -^- x< Ua/'-L ^ _*!_ \1

16L\ 3 3.5 \3 3.5 3.5.7 "/ \3.5 3.5.7 3.5.7.9 "/_]'

*!_ Ue/'-L ^ ^4 \

.5.7 "/ \3.5 3.5.7 3.5.7.9 "/

iJ_JL ^ g* \1

V3.5.7 3.5.7.9 3.5.7.9.11 '"/J

_ --
16L\3 3.5 3

i-x+-.---

Tell1 h3 3.5^ 335 'V 3.5 3.5.7

Now we have

and therefore the three series are respectively equal to

16 _

2 D 2

204 PROFESSOR A. E. H. LOVE ON THE

On substituting for the three series in the expression for A2 we find

A -Aa'[J_ -H-- • --
• 1

L7 45 / 45

. ^ _
rcr s4a4 \ «na*

30. Again we have, when h* = 0 and P = 0,

(

J

5a aa
and with the values already used for fa, . . . this gives

Ari-^4

fe 1
5)"j

-
5vL6 2.7 2.4.7.9 2. 4 . 6 . 7 . 9...(2ic+5)

8AT1 8s3«a S.lOsV , y(2K+2)(2/c + 4)(2ic+6)^a2' 1

^~L7 2.7.9 2.4.7.9. 11 ~ 2. 4. 6. 7. 9...(2« + 7) "J

~ 5v\_ 6.7 2.7.9 2.4.7.9.11 2 . 4. 6 . 7 . 9...(2*+7)

The first of these series has already been transformed into

IG^LV s2**3 s4a*J Jo \saaa s*ct*/j

By means of the identity
(2/c+2)(2*+4)(2>f+6) = i L_ 9 15

(2/c+3)(2ff+5)(2ff+7)" 2/c + 3 (2*c+3)(2»c+5) (2K+3) (2
we transform the series

into

Ifi %y? (-Y _(2^±2)_(2K±4)^_

s|_7 2.7.9 ' 2.4.6.7.9...(2ic+7)"j

l\'/3?x*-\/'Lx> x4 /I x* a;4

16Li1~^+^~''V 3\3"O+3T5^~"7+9l3T5~3^7 + 3.5.7.9~"7

.,

V3.5.7 3.5.7.9 3. 5. 7. 9. 11
which is the same as

GRAVITATIONAL STABILITY OF THE EARTH. 205

and by means of the identity

+ 4) 10 51 120

(2*+3)(2K+5)(2K + 7)~ 2* + 3 (2/c+3)(2ic + 5) (2K+3)(2K+5)(2«+7)
we ti'ansfonn the series

If _j_ tf 8.8s4 , y (2<c-l)(2*+2)(2*+4)xi- "1

5|_~ 6.7 2.7.9 2.4.7.9.11" 2.4.6.7.9...(2»c+7)

into

3.5.9

which is the same as

16
Hence we have

Again we find

(92)

and

»0 1 9 "™" If f9 f\ \ /

Hence the equation lim»,0(A»D/--'-BsC^; 2) = 0 is

-0' • (94)

where x is written for j»a. A factor v~ls~* has been omitted in forming the equation,
as neither v nor * is supposed to vanish. The terms of lowest degree in the left-hand

206 PROFESSOR A. E. H. LOVE ON THE

member of the equation (94) will be found to be •j80i/(24i/— 19), which is negative
when f > v > 0. Hence the left-hand member of this equation is negative when
x = 0. Also it will be found by the method of asymptotic expansion (cf. § 22 above)
that, when x is great, the left-hand member is approximately equal to — 1280aT4 for
all values of v. After the previous cases, in which the corresponding equations have
a single root, we are led to expect that in this case there is no root, for it is unlikely
that there is more than one. We proceed to verify this expectation in the cases
where v = 0 and v = £.

31. Multiply the left-hand member of (94) by a;4 and put v = 0. We get

8\ (3xt+25x2+ 2±Q) -
L

= 0,

or, since 10 is a factor,

9xt+20xa-105-(9x*+Ux4-l5x?-l05)x-le-^{'!el'*dx = 0.

Jo

The term of lowest degree in the left-hand member, when expanded in powers of
x, is — -3fia;8 ; when x is great, the left-hand member approximates to —128. Now
multiply by xe**, and put

y = (Qxi+20x3-l05x)e^-(9x6+llx4-15xa-l05)\Ze^dx.

Jo

We know that when x is small y is small and negative of the order —™x7, and that
when x is great y is great and negative of the order — 128x6**. Now

[*

and, if we put z for x~l dy/dx,

z = (54a?-3Qx)e**- (54z4+ 44z2-30) f'

Jo

where z is negative both when x is small and when x is great ; also

and if we put w for x~l dz/dx,

w = 88a:e'I°-(216z3+88) F tfdx,

Jo
where w is negative both when x is small and when x is great ; and now

</./;

GRAVITATIONAL STABILITY OF THE EARTH. 207

which is always negative. Hence w is always negative, and therefore dz/dx is always
negative, therefore also z is always negative and dy/dx is always negative, and
therefore y is always negative. Thus the equation y = 0 has no real root other than
the irrelevant root x = 0, which was introduced in the process of forming the

equation.

Again, when v = ±, we multiply the left-hand member of (94) by x" and obtain the

equation

(23x9+ 128x4-70xa+ 3675) - (23x"+ 105xs+ 268x4+ 1 155x> + 3675) arVr**1 \^*dx = 0,

of which the left-hand member is of the order -x6 when x is small and -x3 when x
is great. We put
y = (23xT + 128x'-70.r3+3675*)e^-(23x'+105x6+268x4+1155xs+3675)

and then we put

\_dy_ _ Idz^ u_ Idw

and find

— = -(512z4 + 4552xa+3640)e»*t-8832

1 1 .1

which is always negative. Just as before, we deduce that y is always negative.
It is therefore proved that the equation

lias no real root.

32. When n = 3, and h* = 0 and P = 0, we have

A, = (/,
where

«V. *W , y ^a"

— j- •*— -..4- ) 27T^c"V'
-A/. M. *v / y ^q"

"-7"v1"Tfr4~-( } rr^-/'

/ 4A/ a' «»a« <*a' / y_ ^•>«"

s= 77V7" + 2T9~2~rTlH ;2.4...2ic.(2K+5)

A
?

3A

/ y+i _ f"afc w - CT...V
2.4...2*(2K+5)(2K+7) /

208 PROFESSOR A. E. H. LOVE ON THE

We use the identity

(2K+5)(2*+7) 2»c+5 2K+7
to transform Q'3 and then we use the results

7 2.9 2 4

V _ , v_T [

4.11 Jo

We thus find
A., = -

+ ve-*"+?j.v(sa)-1fafe-»dsK\. . (95)
Again we have, when h3 = 0 and k2 = 0,

and here we have

a^Qj_0/ . _3AT J_ *c?

o<« 7^ L 7.9 2.9.11 2.4.11.13 " 2.4...2ie(2jc+7)(2ie + 9)"j'

while the other series can be obtained from those written above. We use the
identity

2/c-l 4 5

2<c+9

to transform the series last written, and we use also the results which we used in
obtaining the expression (95) for A3, and find

63 = ~ [l2(sa)-7^x*e-VJx-15(sa)-9^x8e-^dx+Wv(sa)-^Mx8e-Vdxl . . (96)
We find also the results

GRAVITATIONAL STABILITY OF THE EARTH.
Hence the equation

) = 0
becomes

(99)

27 L ^° ^° J

where x is written for sa.

33. An irrelevant factor v has been introduced into the left-hand member of (99).
We find that when x = 0 this expression becomes

which is positive for all admissible values of v. We find also, by the method of
asymptotic expansion, that the expression is positive, when x is great, for all values
of i/. We proceed to show that, in the important cases v = 0 and v = £, the equation
has no real root. The left-hand member of (99) being an even function of JT, we may
treat x as positive.

When v =«» 0, the left-hand member of equation (99) is

which is positive for small values of x. The differential coefficient of the expression
within the square brackets is

1 4* xV1** dx + 5zV»*,

o

which is positive for all real values of x. Hence the left-hand member of
equation (99) cannot be negative if x is positive, or the equation has no real root.
When v = J, the left-hand member of equation (99) becomes

**]*5^-* (IOO)

The expression within the square brackets is greater than

(101)

where this expression is obtained from the other by replacing every positive coefficient
by the next smaller integer and every negative coefficient by the next greater

integer.

VOL. CCVH. — A. 2 K

210 PROFESSOR A. K H. LOVE ON THE

Since

' *-'*1 dx,

o

the expression (101), when multiplied by 7, is

(5x'-36xa+ 1 54) \*a*e-** dx+5(a*-S)a*e-** ..... (102)

Also we have

FX

*dx =

f»

Jo

9 9. 11 9.11.13 "/'

and therefore the coefficient of x9e~^ in the expansion of (102) is £ (154-9 x 15), the
coefficient of xW is ^(154-36x11 + 5x99), and the coefficient of x9+3xe,-** for
all values of K which are greater than 1 is

36 4J),

9. ll...(2/c+5)\(2/f+7)(2/c+9)
or

20^+88^+145
9. ll...(2fc+5)'

Hence all the coefficients are positive, the expression (102) is positive, and the left-
hand member of (100) is positive for all positive values of x. Thus, in this case also,
the equation (99) has no real root.

Summary of the Solution of the Mathematical Problem.

34. We have now solved in essentials the mathematical problem of the vibra-
tions of a gravitating sphere, initially homogeneous and in a state of hydrostatic
pressure, and have found the conditions of gravitational instability. We have shown
that, when any normal, or principal, vibration is taking place, the dilatation at
a distance r from the centre is specified by the product of a certain function of r and
a spherical harmonic of positive integral degree. We have shown further that, in
each such mode of vibration, the components of displacement can be expressed in
terms of the same spherical harmonic, and that the radial displacement at a point
distant r from the centre is the product of a function of r and the same harmonic.
We obtained the form of the frequency equation, and the forms of all the functions
which enter into its expression.

We proceeded to investigate the conditions which must hold in order that the
frequency equation may be satisfied by a zero value of the frequency. We showed
that, when such a value is not introduced irrelevantly in the process of forming the
equation, its occurrence points to genuine gravitational instability. We found that
the condition of such instability is the condition that a certain equation, containing
the variable quantity x'a*, or f7ry/ju2/(X + 2/i), may be satisfied by a real positive value

GRAVITATIONAL STABILITY OF THE EARTH. 211

of this quantity. The constant p. denotes the rigidity and X + f/n the modulus of
compression. When the harmonic specifying the vibrations is of zero degree, that
is to say, when the vibrations are radial, we found that the critical value of .fa?
lies between 4 and 5 if v, or /n/(X+2/x), is zero, and that it lies between 3 and 4 if v
is \. In the case of vibrations specified by spherical harmonics of the first degree, we
found that the critical value of «*aa lies between 7 '22 and 8 if v = 0, and it lies
between 6'48 and 7'22 if v — \. In the cases of vibrations specified by spherical
harmonics of the second and third degrees we found that there is no critical value of
s3aa, or that the sphere is stable, in respect of the corresponding types of displacement,
for all values of X + 2/i. It was to be expected that the critical values of s*a' would
increase rapidly as the complexity of the type of vibration, specified by the degree of
the appropriate harmonic, increases ; and we appear to be justified in concluding that
instability cannot occur in respect of displacements specified by spherical harmonics of
any degree higher than the first.

35. The result that the critical values of fa* are lower when v = { than when v = 0
means that a higher value of the constant X+2/^ is required, to secure stability,
when there is considerable rigidity than when there is very little rigidity. This
result accords with general dynamical principles ; for it is a known result, and one
which has been shown to be in accordance with such principles, that the frequency of
any mode of vibration, involving compression, of a sphere free from gravitation
diminishes as (X + 2/i)//x, diminishes, that is, as v increases.* Consequently, for a given
value of y/juV, the value of yp0ta?f(\+2p.) which would be required, in order to reduce
the frequency to zero, diminishes as v increases, or the critical value of X+2/x increases
as v increases.

36. The result that the critical value of sW is lower when n = 0 than when n= I
means that a higher value of X+2/t would be required, to secure stability, in the case
of radial displacements than in the case of displacements specified by spherical
harmonics of the first degree. The spherical body of uniform density could l>e stable
in respect of all types of displacement except radial displacements. If the value of
sV were intermediate between the critical values corresponding to n = 0 and n = 1,
this would be the case ; and the body would tend to take up a different configuration,
in which the density would be more concentrated towards the centre. The result
that, in the case where n = 1 also, there exists a critical value of «*«', which
is not more than twice as great as the value associated with n = 0, the initial
state in both cases being one of uniform density, suggests very strongly that there
would be a critical value of X + 2/i, in respect of the case n = 1, even if the configuration
were such that the body was stable as regards radial displacements. We should then
have a body with a spherically symmetrical distribution of density, but with elasticity
too small for this configuration to be stable in respect of displacements specified by
spherical harmonics of the first degree ; and it may be inferred that the critical mean

* Cf. H. LAMB, loc. at., ante, p. 173.
2 E 2

•212 PROFESSOR A. E. H. LOVE ON THE

value of X+2/A for such a body would not be very different from the critical value
obtained for X+2/i by treating the body as homogeneous, and paying attention to
those types of displacement only which are specified by spherical harmonics of the
first degree.

37. If this conclusion is admitted, as I think it must be, it would follow that a
spherical planet with a spherically symmeti-ical distribution of density, and stable as
regards radial displacements, might be unstable as regards displacements of the type
in question ; and then it would tend to be displaced in such a way that the boundary,
or any concentric sphere, moves to a position in which its centre no longer coincides
with the centre of gravity, while the matter in a thin spherical layer becomes
condensed in one hemisphere and rarefied in the other. The density being in excess
in one hemisphere and in defect in the other, and the excess or defect at any point,
at a stated distance from the centre, being proportional to the distance of the point
from the bounding plane of the two hemispheres, the distribution of density may be
aptly described as " hemispherical," and the state of the body may be described as one
of " lateral disturbance." The concentration of density towards one radius, on which
the centre of gravity lies, has the effect of diminishing the potential energy of
gravitation, and this diminution may more than counterbalance the increment of
potential energy due to strain. The proved existence of a critical value for X + 2/A (in
the case of a homogeneous body) indicates that this state of things really can occur.
An illustration of the nature of a hemispherical distribution of density will be found
in §§ 47, 48 below.

38. The results found by JEANS (1903) in the solution of the problem of the
gravitating sphere subjected to an external field of force, which balances gravitation
throughout the sphere when it is at rest, may be compared with those obtained above
in the case where the gravitation is balanced by initial pressure. In JEANS' solution,
just as here, the modes of vibration are specified by the spherical harmonics which
enter into the expression for the dilatation ; and, in any normal mode, the formula
for the dilatation contains a single spherical harmonic, and the radial displacement
at any stated distance from the centre is proportional to the same harmonic. If the
degree of the harmonic exceeds zero, instability can occur for a sufficiently small
value of the resistance to compression, whatever the degree of the harmonic may be.
It is not restricted to the case where the degree is unity, as it is in our problem of
initial stress ; but the value of the resistance to compression required for instability
diminishes rapidly as the degree of the harmonic increases. Instability enters first
when the harmonic is of the first degree,* that is to say, for lateral disturbances.
The critical values of (Pa? are 6 '72 when v = 0 and 5 '33 when v = \, the degree of
the harmonic being unity. Since these values are a little less than the critical values
found in the solution of the problem of initial stress, it may be concluded that the
effect of initial stress, as compared with that of an external field of force, is to

* The question of radial instability was not considered by JEANS.

GRAVITATIONAL STABILITY OF THE EARTH.

213

increase slightly the stability of the body in respect of disturbances specified by
harmonics of the first degree, and to increase it enormously in respect of disturbances
specified by harmonics of higher degrees.

.\l>i'l'"''ili'>i' I" '/<•' I'l'olil' in <>f lln- Hi-'iril'ili<,,,>il St'ili,/,'i/ <>f tli- /•.'"/•'/'.

39. For a body of the same size and mass as the Earth, the values of a and pu in
C.G.8. units are 6'37xl08 and 5'53 ; the value of y being 6'65xlO~8, the value of
Jjry/JoV is 3'46x 10". In the following table the first column gives a value of .iV,
the second column gives the corresponding value of \ + 2/x (the body being of the same
size and mass as the Earth), the third and fourth columns give the values of the
corresponding moduluses of compression in the cases where v = 0 and v = £, irrelevant
entries being omitted. These moduluses are denoted by £0 and A-,. The quantities
given in the fifth, sixth, and seventh columns are the moduluses of compression of
steel, glass, and mercury (denoted by k,, kg, km).

«»tt».

A + 2/i.

*

*i.

*V

V

k,n.

—

—

—

—

1-43x10"

—

3

1 • 15 x 10"

—

7-68x10"

4

8-64x10"

8-64x10"

5-76x10"

—

—

5

6-91x10"

6-91x10"

—

—

—

—

—

—

—

—

4-54x10"

6-48

5-33x10"

—

3-57x10"

—

—

7-22

4-79x10"

4-79x10"

3-19x10"

—

8

4-32x10"

4-32x10"

—

—

—

—

—

—

—

—

2-60x10"

According to these results, a homogeneous solid body of the same size and mass as
the Earth, with a modulus of compression as great as that of steel, would have
complete gravitational stability. If the modulus of compression were equal to, or less
than that of glass, the planet would be unstable as regards radial disturbances, and a
concentration of density towards the centre would take place. If the critical value of

214 PROFESSOR A. E. H LOVE ON THE

X+2/i, which was found in the case of lateral disturbances, is assumed to be the
critical mean value of X + 2fi for a planet in which the mass is condensed towards the
centre, then we may say that, if the mean modulus of compression were about equal
to that of glass, and there were very little rigidity, the planet would be unstable as
regards lateral disturbances ; but, if there were considerable rigidity, it would be
stable. If, on the other hand, the mean modulus of compression were decidedly less
than that of glass, though not so small as that of mercury, the planet would be
unstable as regards lateral disturbances, even though it possessed a considerable mean
rigidity.

40. In order to settle the question of the gravitational stability or instability
of the Earth, we must assign the appropriate values to the constants X and p..
Lord KELVIN'S theory of elastic tides in a solid sphere led to the result that the tidal
effective rigidity of the Earth is not less than that of steel. This result suggests
that fi should not be taken to be less than 8'19x 10" C.G.S. units; but, since it was
obtained by treating the Earth as incompressible, it affords no means of determining
the value of A.. JEANS (1903) proposed to deduce the values of X and /u, from the
observed velocities of propagation of earthquake shocks. In a homogeneous elastic
solid body, free from gravitation and initial stress, irrotational waves of dilatation are
propagated with the velocity [(X + 2ju,)//30]i, where /30 is the density, and equivoluminal
waves of distortion are propagated with the velocity [j*/pvj, while waves of a third
type are propagated over the surface with a velocity approximately equal to
(Q"9)[j*/Ptf- When a great earthquake takes place, the disturbance received at a
distance from the source consists of three sets of disturbances : two sets of
" preliminary tremors," and the " main shock." The first set of preliminary tremors
is received at distant places at such times as it would be if it travelled directly through
the Earth with a velocity of about 10 kiloms. per second. The second set of tremors
is propagated apparently in a rather less regular fashion, but the times at which it
can be observed at distant stations are nearly the same as they would be if it travelled
directly through the Earth with a velocity of about 5 kiloms. per second. The main
shock is received at distant places at such times as it would be if it travelled over the
surface of the Earth with a velocity of about 3 kiloms. per second.* The identification
of the three sets of disturbances with the three sets of waves which are theoretically
known seems to be inevitable, and the discrepancy between the ratio of velocities of
equivoluminal and superficial waves and the ratio of velocities of the second set of
tremors and the main shock may be explained by the supposition that, while the
velocity of transmission of these tremors depends upon the mean rigidity of the Earth
as a whole, the velocity of transmission of the main shock depends upon the average

* Reference may be made to a Memoir by R. D. OLDHAM, " On the Propagation of Earthquake Motion
to great distances," London, 'Phil. Trans. Roy. Soc.,' ser. A, 194, 1900, and to the Reports of the
Seismological Committee of the British Association, in particular that published in ' Brit. Assoc. Rep.,'
1902.

ORAVITATIONAL STABILITY OF THE EARTH. 215

rigidity of surface rock. Assuming this explanation, we are led to attribute to surface
rocks an average rigidity approximately equal to GxlO11 C.G.S. units, and to the
Earth as a whole the much higher mean rigidity 1 '38 x 10ia C.G.S. units; further,
since the ratio of velocities of the first and second set of tremors is approximately
2 : 1, we are led to assume for X+2/i the value 5'53x 10" C.G.S. units, and for v, or
/x/(X + 2jt), the value ^. By analogy to the " tidal effective rigidity " we may introduce
the phrases " seismic effective rigidity " and " seismic effective modulus of compression " ;
and the values of these quantities would be 1'38 x 10" and 3'69 x 1013 C.G.S. units
respectively. When the value of X + 2/i for the Earth is taken to be 5'53x 10", the
corresponding value of .<r*a' is 0'625. The results of § 39 appear to warrant the
conclusion that the moduluses of elasticity of the Earth in its present state are
sufficiently great to render a spherically symmetrical configuration completely stable.

41. In obtaining the above values for X + 2/i and p. no account is taken of
gravitation or initial stress, and it is possible that the most appropriate values would
be a little different from those found atx>ve if gravitation and initial stress, to say
nothing of heterogeneity of density, could be taken into account. For this reason,
although a complete solution of the problem of wave-propagation in a gravitating
planet, even when it is regarded as homogeneous, cannot be obtained, the following
argument may not be without value : — The equations of vibratory motion of a
gravitating sphere in a state of initial pressure have been obtained in § 3 alx>ve.
From equations (10) and (11) of § 3 we can deduce the equation

(1Q3)

and the three equations of the type

at* f
where CTT, tsry, or. denote the components of rotation, so that

9 3w ov ,1ft,v

— CT »• ^— "™ "" ~~ * ...... • • • • • • • * I 1 v *J I

oy oz

In a general way we can see that the terms which contain «* in these equations are
small compared with the remaining terms ; for, if waves of length L are propagated,
V3A is of the order Lr*A, and s*A is small in comparison with this in the order «*!/,
which is comparable with La/a', since **a* is comparable with unity. It would thus
appear that the velocities of propagation of the waves are not much affected by
gravitation and initial stress when the wave-length is small compared with the radius
of the sphere ; and the conclusion would be applicable to superficial waves as well as
to waves of dilatation and waves of distortion, because such waves are, in any case,
to be investigated by means of equations of the types (103) and (104).

210 PROFESSOR A. E. H. LOVE ON THE

42. In the case of waves of dilatation the argument can lie put in a more definite
shape. Let us suppose that, near a place, the waves are plane, so that A is a function
of a; and t, and let us write

(106)

so that Vj is the velocity of waves of dilatation when gravitation and initial stress
are disregarded. We have the equation

or

In considering the passage of waves near a place, we may treat the term — ^s*y? in
the coefficient of e1>!zQA as a constant ; and then the equation is satisfied by putting

= B cos {2irL-l(x-xl)- V',*)},
provided that

Since the greatest values of sV are comparable with unity, the value of V\, the
local velocity of transmission, is a little less than VI( or the actual value of X + 2/x is a
little greater than the seismic effective value. The result (107) may be accepted as
being not far from the truth in a region large compared with the wave-length, and
small compared with the radius, and situated at a considerable distance from the
source of disturbance.

43. Since the equations of type (104) contain the dilatation as well as the
components of rotation, it appears that the customary law of independence of waves
of dilatation and waves of distortion ceases to hold when gravitation and initial stress
are taken into account. It appears also that the velocities of propagation, both oi
those waves which are mainly dilatational and of those which are mainly distortional,
depend on the wave-lengths, and, for the same wave-length, they vary from place to
place. When the theory can be developed further, these results may possibly prove
to be Useful in explaining the observed irregularities in the propagation of the
tremors which are recorded in the case of great earthquakes. The high values which
seismic observations lead us to attribute to the elastic constants of the earth as a
whole are in accord with Lord RAYLEIGH'S view* that great initial stress increases

the effective values both of resistance to compression and of rigidity.

i i '

* Lof. fit., ante, p. 173.

ORAVITATIONAL STABILITY OF THE EARTH. 217

PART II.

A Past State of Gravitational Instability as a Reason for the existing Distribution

of Land and Water.

44. Although the conclusion reached by JEANS (1903), that a spherical planet of
the same size, mass and elasticity as the Earth, in its present state, would be in
a condition of gravitational stability, is confirmed and strengthened by the present
investigation, it by no means follows that the Earth has always been in such a state
as it is now. The fact that the mean density of the Earth as a whole is greater than
the average density of surface rocks points to a concentration of mass towards the
centre, and suggests that such a concentration may have come about through the
elasticity having once been too small for a homogeneous state to be stable. We have
seen that this would have been the case if the modulus of compression was once as
small as, or smaller than, that of glass. But we also saw reason to think that, if the
mean modulus of compression was once decidedly less than that of glass, spherically
symmetrical states of aggregation would also have been unstable, and the body would
have existed in some other state. Further, we saw that, if the body was at rest, the
state in which it would have existed is that which we have described as a state of
lateral disturbance with a hemispherical distribution of density. The excess of
density in one hemisphere and defect in the antipodal hemisphere would have existed
alongside of the concentration of mass towards the centre.

45. In the paper already cited JEANS (1903) struck out the idea that the
distribution of land and water on the surface of the globe is associated with a past
state of gravitational instability. He had found that such instability would manifest
itself in what has been called above a hemispherical distribution of density. When
the square of the irregularity is neglected, the figure of a planet at rest, with such
a distribution of density, is a sphere, but the centre of figure does not coincide with
the centre of gravity. On taking account of the square of the irregularity, JEANS
found that the surface of the planet, still supposed to be at rest, would be such as
can be described roughly as a nearly spherical ellipsoid of revolution, with one half
slightly flattened at the middle, and the other slightly tapered in the antipodal
direction. The figure was described as " pear-shaped," the " pear " having a blunt
end, a sharper end, and a waist. The waters of the ocean would presumably collect
in the hollow of the waist, and JEANS pointed out that there is some resemblance of
the shape of the Earth to this figure, although the " stalk " end of the "pear " was
difficult to discover.

In the same year a paper was published by W. J. SOLLAS,* in which it was
concluded from a discussion of the geographical facts that the shape of the Earth

* " The Figure of the Earth," ' Quart. J. Geol. Soc.,' 59 (1903), p. 180.
VOL. CCVII. — A. 2 F

218 PROFESSOR A. E. H. LOVE ON THE

resembles that of a "pear"; but SOLLAS' and JEANS' "pears" have little in common
beyond the name. JEANS' ideal distribution would consist of a hemisphere which is
nearly all land, and an antipodal hemisphere which is nearly all ocean, with a central
island in the middle of this ocean. SOLLAS' account of the actual distribution is that
in one hemisphere there is a central continent (Africa) nearly surrounded by a belt of
seas, while in the antipodal hemisphere there is a central ocean (the Pacific) nearly
surrounded by a ring of land, the belt and ring being broken at three places, which
are distributed nearly symmetrically around the centres of the two hemispheres.
This description suggests very strongly a mathematical account expressed in terms of
surface harmonics of the third degree.

If we neglect the rotation of the planet, and regard it as at rest under no external
forces, we can reach no other result than that reached by JEANS, viz., that, if the
modulus of compression was once so small that a spherically symmetrical state of
aggregation would have been unstable, the state of the planet would have been one
of lateral disturbance with a hemispherical distribution of density. We should not
be in a position to account at all for the geographical facts as presented by SOLLAS.

46. The Earth is a rotating globe, and it is now generally believed to be the
larger of two fragments into which a single body has been broken up ; the other
fragment is the Moon. In the early history of the Earth-Moon system the two
fragments rotated, nearly as a single rigid body ; the period of revolution of the
Moon was nearly the same as the period of rotation of the Earth. We wish to trace
the consequences of supposing that the average elasticity of the material was once
much smaller than it is at present— that the average modulus of compression was
more of the order of that of mercury, or even water, than of that of glass or steel,
and the average rigidity was" smaller in comparison with the modulus of compression
than it is to-day. We have the problem of determining the distribution of density
within the planet, and the consequent shape of its surface. The problem cannot be
solved completely, but we can make some progress with it ; and we can then attempt
to discover the extent to which our results accord with geographical observation. In
so far as the accord is good we may regard geography as supporting the hypothesis
as to the past state of the Earth.

Illustration of the Nature of a Hemispherical Distribution of Density.

47. We have reason to think that, in the absence of rotation and external forces,
the planet, if of sufficiently small elasticity, would have been in the state which we
have described as a state of lateral disturbance with a hemispherical distribution of
density. Before proceeding to take account of the rotation and external attractions,
we consider further the nature of such a disturbance. For this purpose we take the
problem of a spherical body, homogeneous when unstrained, and devoid of all rigidity,
and suppose that in the initial state the self-attraction of the body is balanced by

GRAVITATIONAL STABILITY OF THE EARTH. 219

hydrostatic pressure. We suppose also that the law of elasticity of the body is that
the increment of pressure is proportional to the increment of density. We show
that equilibrium is possible in strained states, in which the excess of density at any
assigned distance from the centre is proportional to a spherical surface harmonic of
the first degree.

In the initial state the pressure pa and potential V0 are given by the formulae

Po = |T>Y>.' (a'-S), V0 = firypo (3a'-r»).

In the strained state the pressure P, density p, and potential V are expressed by
the formulae

where £ denotes the condensation. The equations of equilibrium are

av ap av ap av ap

'-" 0) p-= °' ?--'-

and W is connected with £ by the equation

When terms of the second order in the small quantity f are neglected, the
equations of equilibrium become three equations of the type

-x-Oi ....... <108>

and, on eliminating W, and writing s* for $7ry/>02/X, we have

V'£+sV |£ + 6^=0 ......... (109)

or

This equation is satisfied by putting

where A is an arbitrary constant and <u, is a spherical solid harmonic of the first
degree, and this is the most general form of solution in which f is finite at r = 0, and
is proportional when r = const, to a surface harmonic of the first degree. The
additional potential W has the form

W = ^^{lAa-'e-i-X+F,},

where FI denotes a spherical solid harmonic of the first degree.
Let the bounding surface become

r = a + U..
2 F 2

220 PROFESSOR A. E. H. LOVE ON THE

Since the pressure vanishes at this surface, the expression

Ivyp* {a3- (a + U0)a} + XA ( 1 -i«V) e~^a (ujr)
vanishes, or we have, neglecting Ua2,

so that Ua contains the same surface harmonic as a,. The form of Fj is determined
by the condition that W is the potential of a distribution of density p0£ through the
volume of the sphere r = a, together with a distribution of density p0Un on its
surface. Just as in § 14, this condition leads to the equation

F, = tVAs-V^V

48. Now let the bounding surface in the strained state be

r = a + b cos 0,

which represents a sphere with its centre at a small distance b from the origin in the
direction of the axis of the harmonic. We find

r cos ,

W = -

S ft *~ 0

br cos 6.

If s2a2 > 5, the condensation is greatest near the centre, and it is positive on the
side remote from that towards which the surface is displaced, so that the centre of
gravity is displaced in the opposite sense to the surface. The distance of the centre
of gravity from the origin is easily proved to be 5b/(s2a2— 5).

Fig. 1.

Fig. 2.

GRAVITATIONAL STABILITY OF THE EARTH. 221

The variation of the excess density along the axis of the harmonic is illustrated in
fig. 1. The surface r = a(l + ecos0) can be an equipotential surface if

ne = -

and thus a sphere of radius a with its centre at the displaced centre of gravity is an
equipotential surface. The relative situation of the bounding surface and of this
equipotential is illustrated in fig. 2, in which O denotes the undisplaced centre, C the
centre of the displaced surface, and G the centre of gravity of the strained sphere.
The figures are drawn for the case in which s*a* =10.

The type of disturbance which has been called above a lateral disturbance with a
hemispherical distribution of density would be the same in a body possessing some
degree of rigidity, but the numerical details would be different.

49. If the equipotential surfaces of a nearly spherical body, with a nearly
symmetrical distribution of density, are referred to the centre of gravity of the body
as origin, their equations take such forms as

r =

in which « a, ... denote small coefficients, and S_.. . . . denote spherical surface harmonics
of degrees indicated by the suffixes. There is no term of the form e,S,. In the case
of the Earth, the coefficients c2, ... can be determined by means of pendulum
experiments. If we referred to a different origin, near the centre of gravity, a term
of the form €,8! would be introduced, but the coefficient e, could not be determined by
means of pendulum experiments, for it does not affect the formula for the variation
of gravity over the surface.* If we choose an origin in accordance with geometrical
considerations, e.g., as the centre of that oblate spheroid which most nearly coincides
with the surface of the ocean, the results of pendulum experiments cannot tell us
whether this origin coincides with the centre of gravity or not.

Effect of Rotation upon a Planet with a Hemispherical Distribution of Density

50. In all the preceding work the rotation of the Earth has been neglected. We
have now to consider the effect of rotation upon a nearly spherical planet which, in
the absence of rotation, would have a hemispherical distribution of density. To
simplify the analysis, we shall disregard the concentration of mass towards the
centre and also the rigidity of the body. We shall take as the " initial " state of the
body a state in which the density is uniform and the stress is hydrostatic pressure,

* The result may be inferred from STOKES' investigation of the " Variation of Gravity over the
Surface of the Earth," Cambridge, ' Trans. Phil. Soc.,' 8 (1849), or ' Math, and Phys. Papers,' vol 2,
Cambridge, 1883. It is easy to prove it independently.

222 PROFESSOR A. E. H. LOVE ON THE

while the body is rotating, as if rigid, about the axis of z with angular velocity a> ;
and we shall seek a strained state in which the body could exist without the
application of any external forces, this state being such that, in the absence of
rotation, the distribution of density would be hemispherical. In the notation of § 47
the equations of steady motion of the body are

av ap av ap av ap M

-P<JX = P---, -p(ay = p—--, "P^-fc. • . (HO)

The initial state is determined by the same equations with p0, V0, £>0 substituted for
p, V, P. Now the initial figure is an oblate spheroid, and the initial form of V is

V0 = const. -£ {A'
where A' and C' are constants ; also the initial form of P is

p0 = const. +/>0 |V0

= const, -ifl, {(A'-o,2

When we write, as in § 47,

and neglect terms which cancel on account of the values of p0 and V0, and
also neglect terms which are of the second order in the small quantity £ the
equations (110) become

(111)

3x 82: '
8W_X8|

8W_X§£

Now we have the equations

2A' + C' = 4 Try/ao, V2W = — 4 777/3,,
and therefore we can eliminate W and obtain the equation

where s" is written for f 7ryp02/A. If w were zero, A' and C' would both be equal to
sTryp,,, and we therefore put

A", C' =

GRAVITATIONAL STABILITY OF THE EARTH. -J23

then equation (112) becomes

+y+C». . (us,

The left-hand member of this equation is the same as that of equation (107) in § 47
above ; and therefore, when to* is neglected, f can be of the form &, where

the notation being the same as in § 48. Now we shall suppose that o>* is not large,
so that we may treat £, as an approximation to £, and substitute & for f in the
right-hand member of equation (113), for all the terms of this member are small of
the order «*£ We are then neglecting f*, but not w*£ To obtain a second
approximation, we put

f-

and seek a particular integral of the equation

There would be no special difficulty in obtaining a solution of the equation, but it
will be sufficient for our purpose to find the form of the solution. The function £,
may be expressed in terms of polar co-ordinates r, 0, <f> in the form

f, —f(r) r (a, sin 0 cos <f> + /3i sin 0 sin <£ + yi cos 0),

where a,, /?,, y, are constants, and ./'('') is a certain function of r which has been
determined. Hence we have

r/(r)(«, sin 0 cos <j>+ 0, sin 0 sin <£)
ox cy

+ i*f (r) sin2 0 («i sin ^ cos <j> + /8, sin ^ sin <f> + yl cos 0),

z > = r/(r) y, cos ^+ r»/' (r) cos* 0 (a, sin 0 cos ^+0, sin 0 sin <£ + yi cos 0) ;

oz

and these can be expressed in the forms

8n +''r

-/-»/' (r)(coss 0-|) sin 0(a, cos <£+/8, sin «^) -ry'(r) yt (cos3 0- f cos 0),

+ r*/'(r) sin 0(cos3 0~i)(«, cos <£+& sin <^) +ry(r)yi (cos3 6-$ cos

Hence the right-hand member of (114) can be expressed as a sum of terms each of
which is the product of a function of r and a spherical surface harmonic, and the

224 PROFESSOR A. E. H. LOVE ON THE

surface harmonics which occur are those of the first degree and the following
harmonics of the third degree :—

cos3 6— f cos 0, (cos2 0— £) sin 9 cos <j>, (cos2 0— ^) sin 0 sin <f>.

To each of these terms there corresponds a term of the same form in £', and
therefore also in P, or p0+\£; and it follows that the displacement of the bounding
surface from its initial form (which is a slightly elliptic oblate spheroid appropriate to
the rotation) is expressed by a radial displacement, which consists of a part propor-
tional to a spherical surface harmonic of the first degree, together with parts propor-
tional to the above surface harmonics of the third degree. In like manner all the
terms of the additional potential W are the products of functions of r and surface
harmonics, which are either of the first degree or are the above harmonics of the
third degree ; but the coefficients of the various harmonics in W are different from
their coefficients in £ The equation of the equipotentials under gravity, modified by
the rotation, is

Ve + W + £<o2 (x? + y3) = const. , or p0/pa + W = const. ;

and thus the situation of the bounding surface relative to the equipotentials is
expressed by a difference of radii at corresponding points, this difference being a sum
of terms of the form &S, where b denotes a constant and S denotes a surface
harmonic ; and the surface harmonics which can occur are those of the first degree
and the three of the third degree written above.

51. It appears from this investigation that, if a gravitating body, which is rotating
about an axis, has so small a modulus of compression that, if the body were at rest, a
spherically symmetrical distribution of density would be unstable, it would tend to take
up a state in which the distribution of density would not be exactly hemispherical,
but the excess density would also contain terms expressed by spherical harmonics of
the third degree. The figure of the body would differ from the oblate spheroidal
figure appropriate to the rotation by a radial displacement at each point ; and this
displacement would be expressed partly by spherical surface harmonics of the first
degree, indicating that the centre of gravity does not coincide with the centre of
figure, and partly by spherical harmonics of the third degree. If the body were
entirely devoid of rigidity, the oblate figure appropriate to the rotation would be the
same as that of an equipotential surface under gravity, modified by the rotation ; and
the figure of the body, as determined by difference of level above or below a certain
equipotential surface, would be an harmonic spheroid of the third degree, and the
situation of the body would be that of such a spheroid when displaced towards one
side. If the body possessed some rigidity, the oblate figure appropriate to the
rotation would differ a little from that of a nearly coincident equipotential surface,
and the shape of it, determined as before, would be that derived from a certain oblate
spheroid of small ellipticity by a displacement proportional to a surface harmonic of
the third degree. The surface harmonic would be of a somewhat specialised type.

GRAVITATIONAL STABILITY OF THE EARTH. 225

Effect of certain External Forces.

52. The effect of forces such as the attraction of the Moon at the time when its
period of orbital revolution did not differ much from the period of rotation of the
Earth would l>e to draw the planet out into a shape more nearly ellipsoidal, with
three unequal axes, than spheroidal. If the planet could have had a symmetrical
shape it would have l)een practically ellipsoidal, and the surfaces of equal density
would have been ellipsoids. Whereas the effect of rotation is the same as that of
forces derived from a potential of the second degree, symmetrical ivlxmt the axis of
rotation ; such forces as we are now considering are derived from a potential, the
most important terms of which would lx? of the second degree, but not symmetrical
about the axis of rotation. If the elasticity was too small for an ellipsoidal figure to
l)e stable, the planet would have been in a disturbed state, the nature of which can
1)6 inferred from the preceding investigation. We have only to replace in § 50 the
initial potential, modified by the rotation, by a general expression of the second degree
in the co-ordinates. The only change that would be made in the result would be that
those terms in the radial displacement which are expreased by harmonics of the third
degree would not be of the specialised type introduced by the rotation, but would be
of general type. The figure of the planet would be derived from the ellipsoidal
figure appropriate to the rotation, and to the external forces, by a radial inequality
expressed by sin-face harmonics of the first and third degrees. The equipotential
surfaces would IK? obtained from the ellipsoidal equipotentials appropriate to gravity,
modified by the rotation and the external forces, by surface harmonics of the same
degrees. The result would be that the shape of the planet, as determined by
difference of level above or below a certain equipotential, would be a wrinkled
ellipsoid, displaced towards one side ; and the wrinkle would be expressible by means
of a spherical surface harmonic of the third degree.

Tlie Problem of the Shape- of the Lithosphere.

53. The problem of determining the form of the equipotentials near the surface of
the Earth includes the problem of determining the figure of the surface of the
ocean (the " hydrosphere "). The equipotentials which lie outside the nucleus (or
" lithosphere ") on one side, and sufficiently near to it, cut the surface of the
lithosphere towards the other side. Among these equipotential surfaces that one
which, outside the lithosphere, coincides with the surface of the ocean is known as
the " geoid." The surface of that part of the lithosphere which lies outside the geoid
is occupied by land, and can be observed directly ; the surface of that part which lies
within the geoid can only be observed indirectly by means of soundings. We have
no means of investigating the form of the surface of this part of the lithosphere
except by estimating its depth at a point below the geoid. The most important
deviations from sphericity both of the lithosphere and of the geoid are of such

VOL. CCVII. — A, 2 O

226 PROFESSOR A. E. H. LOVE ON THE

a nature that these surfaces are nearly oblate spheroids. If the lithosphere were
exactly iu the form of an oblate spheroid, and its centre of gravity coincided with its
centre of figure, it would either lie entirely within the geoid or would protrude from
it symmetrically at the North and South Poles. Owing to the rigidity of the
lithosphere the ellipticity produced in the geoid by rotation would be slightly greater
than that produced in the lithosphere, and thus there is a tendency to lay bare the
polar regions ; but, since the land of the globe does not consist of two circular islands
at the poles, there are other deviations from sphericity, both of lithosphere and geoid,
and the relative amounts of these at different places can be expressed by the difference
of radii drawn from the centre of gravity. According to the theory which has been
here advanced this difference of radii should be, at least in its general features,
expressible as a sum of spherical harmonics of the first, second and third degrees.

54. It is easy to verify the presence of some of these harmonics. The effect of
a term of the first degree would be to make the lithosphere protrude from the geoid
towards one side. If this term were the only one, the land of the globe would form
a circular island or continent. It is the fact that most of the land is in one
hemisphere. The great circle of the globe which contains most land has a pole
situated between Orleans and Le Mans* (latitude 48° N., longitude 30' E.). Again,
the zonal harmonic of the third degree vanishes at three circles, one being a great
circle. If this term were the only one, the land of the globe would consist of
a circular island surrounded by a belt of ocean in one hemisphere, and in the
antipodal hemisphere there would be a circular ocean surrounded by a ring of land.
This arrangement corresponds to two features of SOLLAS' description of the Earth's
surface. The nearly symmetrical breaking at three places of the belt and three of
the ring, which he also noticed, indicates the presence of the sectorial harmonic of the
third degree. If we refer to the polar axis, instead of any other morphological axis,
the presence of the zonal harmonic of the third degree is indicated by the existence
of an Antarctic continent, and by the fact that most of the land of the globe is north
of the Equator. The harmonic of the third degree and second rank, referred to the
polar axis, vanishes at the Equator and at four meridians symmetrically placed. If
this term were the only one, then, in two northern quadrants there would be land,
and also in the two alternate southern quadrants, an arrangement which suggests
Central Asia and North America as the land quadrants of the northern hemisphere,
Australia and South America as those of the southern.

Spherical Harmonic Analysis of tlie Distribution of Land and Water.

55. By such arguments as the foregoing, and by some trials with small numerical
coefficients for the various harmonics, I had convinced myself that many features of
the distribution of land and water could be represented by means of harmonics of the
third degree, when Professor H. H. TURNER suggested to me the advisability of

* E. BRUCKNER, 'Die feste Erdrinde und ihre Formen,' Wien, 1897.

GRAVITATIONAL STABILITY OF THE EARTIL 227

adopting a systematic process for the discovery of appropriate coefficients. He very
kindly made, and placed at my disposal, a rough preliminary calculation, and the
results were sufficiently encouraging to warrant the undertaking of a considerable
piece of computation. A professional computer was employed for a time, but
eventually I relied upon my own calculations, taking many precautions to ensure
accuracy. The systematic process consists in devising a function to represent the
" value of land " at any point, and determining, by the method of approximate
quadrature, the coefficients of an expansion of the function in spherical harmonics.
The results of such a computation clearly depend upon the chosen " value of land,"
and judgment must l>e exercised in selecting appropriate values. Little importance
can be attached to the heights of mountains, l>ecause the highest mountain ranges
are, geologically speaking, modern, the ancient mountains being worn down by
denudation and erosion. Too much importance is not to be attributed to the actual
coast-line, localise this line is subject to many causes of change. The coast-line is
but one of the contour-lines of the continental block (the geoid being the level of
reckoning), and the shaj>e of the block at considerable depths differs a good deal from
that at the surface. At mean-sphere-level (8400 feet below sea-level) the continents,
with the exception of the Antarctic continent, form a continuous block.* The Arctic
Ocean is reduced, so far as is known, to a trough running nearly along the meridian
of Greenwich, from about latitude 05° N. to about latitude 80° N. It may extend to
the North Pole and surround it. The polar block spreads southwards in two great
masses — America and Eurasia. These are joined through the British Isles, Iceland
and Greenland on the one side, and across Behring's Strait on the other ; the contour-
line at mean-sphere-level runs practically along the 60th parallel between America
and Europe and along the 50th parallel between America and Asia. The Eurasian
division of the block forks near the Persian Gulf, and tapers southwards in two
branches, one containing Africa and the other the Malay Peninsula, adjacent islands,
Australia and New Zealand. The Red Sea does not go down to mean-sphere-level, and
the Mediterranean does so only in two small patches. The American division of the
block is continuous across the Gulf of Mexico, the West Indies and the Caribl>ean
Sea, which, at this depth, equally with Mexico, Central America, and the Isthmus of
Panama, form part of the ridge joining North and South America. The ridge has
some local depressions. The block tapers towards Cape Horn, in the neighbourhood
of which, however, it has a great eastward extension, and this extension turns
westward and nearly joins the northern continental block to the Antarctic continental
block through the South Shetland Islands.! The Antarctic block also shows a

* The information here detailed in regard to the distribution of the continental blocks and oceanic
regions at mean-sphere-level is taken from a map drawn by H. R. MILL in ' The Scottish Geographical
Magazine ' (Edinburgh), vol. 6 (1890), p. 184. Reference may be made to the rough map on p. 237 below.

t It is now known that the depth of the channel is not so great as it was for a long time supposed to be.
See a paper by W. S. BRUCE in 'The Scottish Geographical Magazine ' (Edinburgh), vol. 21 (1905), p. 402.

2 O 2

228 PROFESSOR A. E. H. LOVE ON THE

northward extension towards Australasia. The contour-line of the continental blocks
at mean-sphere-level is a very important and fairly well ascertained datum of the
problem. If, however, we attend exclusively to it, we are liable to emphasise unduly
those parts of the block which do not rise above the level of the sea.

56. I calculated the coefficients of a spherical harmonic expansion up to harmonics
of the third degree for two different assumptions as to the " value of land." lu the
first assumption the value — 1 was attached to those points of the surface which are
below mean-sphere-level and the value 0 to those points which are above it. In
the second assumption the value 1 was attached to those points of the surface which
are above sea-level and the value 0 to those below it. The coefficients obtained by
the two assumptions were then added. The somewhat greater importance of the
mean sphere may perhaps be sufficiently represented by the result that the maxima
obtained by using the first set of coefficients are larger than those obtained by using
the second set. The combined distribution for the two sets of coefficients is shown in
the following table, in which 6 stands for co-latitude measured from the North Pole,
and <f) for longitude measured eastwards from the meridian of Greenwich :—

GRAVITATIONAL STABILITY OF THE EAItTH.

'2 21)

TABLE.

\ *

\

0".

5°.

10".

15°.

20*.

25°.

30°.

35°.

40°.

45°.

50°.

55'.

60°.

65*.

70°.

75'.

80°.

85°.

» \

•

1

i

1

1

i

1

i

1

1

i

i

i

i

_ i

-- 1

i

. i

i

10

-1

15

-1

20

-1

1

1

1

25

-1

1

1

1

1

1

1

1

1

30

1

1

1

1

1

1

1

1

1

35

1

1

1

1

1

1

1

1

40

1

1

1

1

1

1

1

45

1

1

1

1

I

1

1

50

1

1

1

1

:,:,

1

1

1

1

60

1

1

1

1

1

1

1

65

1

1

1

1

1

1

70

1

1

1

1

-1

1

75

1

1

1

1

1

-1

80

1

1

1

-1

-1

85

1

1

1

-1

-1

-1

-

90

1

1

-1

-1

-1

—

-

-1

-

95

-1

1

1

-1

-1

-1

-1

_

_

-1

—

100

-1

1

1

-1

-1

-1

-1

_

-

_ i

_

105

-1

1

1

1

-1

-1

-1

_

_

-1

_

110

-1

1

1

1

-1

-1

-1

_

-

_ i

-

116

-1

-

1

1

-1

-1

-1

_

_

_ i

—

120

-1

_

-1

—

-1

-1

-1

-1

-1

_

_

-1

_

10K

_ i

_ i

i

i

1

I

i

i

i

1 — • '

130

A

-1

_

^ A

-1

• i
-1

_

_

-1

-1

-1

-1

-1

-1

_

-1

_

135

-1

-

-1

-1

-

_

_

-1

-1

-1

-1

_

-1

-1

_

-

-1

-

140

-1

-

-1

-1

-

-

-

-1

-1

_ i

-1

—

-1

-1

-

-

-1

-

145

-1

-

-1

-1

_

-

_

-1

-1

-1

-1

_

-1

-1

—

_

-1

_

150

-1

-1

-1

-1

—

_

_

-1

-1

-1

-1

_

-1

-1

_

-

-1

_

] :,:,

-1

-1

-1

-1

-1

160

165

1

1

1

170

1

1

1

1

1

1

175

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

PROFESSOR A. E. H. LOVE ON THE

TABLE (continued).

\ <h

\

90'.

95".

100°.

105°.

110°.

115°.

120°.

125°.

130°.

135°.

140°.

145°.

150°.

155°.

160°.

165°.

170°.

175°.

"

5

1

1

1

I

1

i

i

i

1

1

i

i

1

1

i

1

i

1

10

15

1

1

20

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

25

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

30

1

1

1

1

1

1

1

1

1

1

1

35

1

1

1

1

1

1

1

1

1

1

40

1

1

1

1

1

1

1

1

1

1

-1

-1

-1

-1

45

1

1

1

1

1

1

1

1

1

1

-1

-1

-1

-1

-1

50

1

1

1

1

1

1

1

1

-1

-1

-1

-1

-1

-1

55

1

1

1

1

1

1

1

-1

-1

-1

-1

-1

-1

-1

60

1

1

1

1

1

1

1

-1

-1

-1

-1

-1

-1

-1

65

1

1

1

1

1

1

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

70

1

1

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

75

1

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

80

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

85

-1

1

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

90

-1

1

1

-1

-1

-1

-1

-1

-1

-1

-1

95

-1

-1

1

1

1

-1

-1

-1

-1

-1

-1

100

-1

-1

-1

-1

-1

-1

105

-1

-1

-1

-1

-1

-1

1

1

-1

-1

-1

- 1

-1

110

-1

-1

-1

-1

-1

1

1

1

1

1

-1

-1

115

-1

-1

-1

-1

1

1

1

1

1

1

1

1

-1

-1

120

-1

-1

-1

-1

1

1

1

1

1

1

1

1

-1

125

-1

-1

-1

-1

1

1

1

-1

-1

130

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

1

135

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

_

140

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

145

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

_

-1

-1

-1

150

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

_

-1

-1

-1

155

1

1

1

1

1

1

1

1

1

1

1

1

160

1

1

1

1

1

1

1

1

1

1

1

1

165

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

170

1

1

1

1

1

1

1

1

1

1

1

1

j

1

1

1

1

175

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

KKAVITATIONAI, STABILITY OF Till. KAU'll.

•J31

TAHLK (continued).

\ «/>

\

180'.

185".

190J.

195'.

205'.

210.

215a.

220.

225'.

230".

235\

240'.

245".

250.

255".

260.

265*.

(j

i

j

i

i

1

i

i

i

_ i

j

_ i

i

i

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1

10
15
20
25
30
35
40
45
50

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1
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1

1
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1

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1

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1
1
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130
135
140
145
150
155
160
165
170
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1

232

PKOFESSOE A. E. H. LOVE ON Till-

TABLE (continued).

\ *

\

270°.

275°.

280°.

285°.

290°.

295°.

300°.

305°.

310°.

315°.

320°.

325°.

330°.

335°.

340°.

345°.

350°.

355'.

9 \

0

5

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1

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75

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80

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85

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90

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95

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110

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115

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1

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1

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120

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1

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1

1

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1

1

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135

-1

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1

1

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-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

140

-1

-1

1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

145

-1

-1

-1

-1

— 1

-1

-1

-1

150

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

155

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

160

1

1

1

1

-1

-1

-1

-1

165

1

1

1

1

1

1

1

1

1

1

170

1

1

1

1

1

1

1

1

1

1

1

1

1

1

175

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

.

GRAVITATIONAL STABILITY OF THE EARTH. 233

57. The surface harmonics of the first degree expressed in ordinary spherical polar

co-ordinates 6, d> are . /, . /, • ,

sin 0 cos <p, sm 0 sin <f>, cos 0 ;

and any spherical surface harmonic of the first degree can be expressed in the form

(p cos<£ + </ sin <f>) sin 0+r cos 0, (115)

where p, q, r are numbers. The spherical surface harmonics of the second degree

' I 1't ' *

sin 20 cos <f>, sin 20 sin <f>, sin*0cos2<£, sin* 0 sin 2<£, 3 cos 20+1;
and any spherical surface harmonic of the second degree can be expressed in the form
(acos<£ + y8sin<£)sin 20+ (y cos 2^ + Ssin 2<£) sin* 0+e(3 cos 20+ 1). . (116)

The spherical surface harmonics of the third degree are
(i.) The zonal harmonic g ^ 0_^ ^ Q .

(ii.) The tesseral harmonics of the first rank

(5 cos2 6— 1 ) sin 0 cos <£, (5 cos3 0— l)sin 0sin <f>;
(iii.) The tesseral harmonics of the second rank

sin" 6 cos 6 cos 2$, sin* 0 cos 0 sin 2<£ ;
(iv.) The sectorial harmonics

sin" 0 cos 3<f>, sin3 0 sin 3<f>.
Since

5 coss 0-3 cos 0 = | (cos 30+ £ cos 0),
(5 cos3 0-1) sin 0 = \ (sin 0+ 5 sin 30),
sin* 0 cos 0 = \ (cos 0- cos 30),

sin8 0 = \ (3 sin 0- sin 30),
any spherical surface harmonic of the third degree can be expressed in the form

aW + X(hcoa<f> + c8in<f>)+Y(dcos2<t> + esin2<f>)+Z(fcos3<l> + (j8m3<l>), . (117)
where n, b, c, d, e, f, g are numbers, and

W = cos 30+ (0-6) cos 0, I
X = sin 0+5 sin 30,

Y = cos 0- cos 30, [ (118)

Z = 3 sin 0- sin 30.

1 The form 3 cos 20 + 1 for the zonal harmonic is 4 (| cos* 0 - J), and is token aa teing more convenient
for calculation.

VOL. CCVII.— A. 2 H

234

PROFESSOR A. E. H. LOVE ON THE

58. Let F(0, <£) denote the function to be expanded. The coefficients are expressed
by equations of the type

p ( " <ty f (16 (cos <£ sin e)2 sin 6 = \ d<f> f d0F(0, <f>) (cos <f> sin 6} sin 0. . (11 9)

J o J o J o J o

The factors multiplying the coefficients p, &c. in the left-hand members are the
integrals of the squares of the several harmonics over the surface of a unit sphere.
The integrals in the right-hand members are the integrals, over the surface of the
same sphere, of the product of the function to be expanded and the corresponding
harmonics. The values of the integrated squares multiplying p, &c., are recorded in

the following table : —

Coefficient.

p,q,r

a, , 7'

d,e

f,9

Value of integrated
square.

1

Reciprocals.

Since the ratios only are relevant, the integrals in the right-hand members of such
equations as (119) are to be multiplied by the numbers in brackets in the third
column.

59. To evaluate integrals of the type in the right-hand member of (119), when the
value of F (0, <f>) is given by the table of § 56, or any similar table, we treat the
integral as a double sum, e.g.,

71

'

30

36

then we have to evaluate such a double sum as

v v V iH7r mir\
« 2, j I — , —

m=o B=I '^oo 30 y

u~

GRAVITATIONAL STABILITY OF THE EARTH. 235

We sum first with respect to m ; but in forming the sum we take account of the
fact that sinJ(n7r/36) does not change when n is replaced by 36 — M. For example,
let F be equal to 1 at the points indicated in the table, and zero at other points.
Then the contribution to the terms containing any m of the two parallels given by n
and 36 — n is either 0, 1, or 2, according as a 1 occurs on neither parallel (for the
particular m in question), on one, or on lx>th. This number 0, 1, or 2 is to be
multiplied by tin- value of cos(mir/3G) for the chosen m ; but the same value for the
cosine occurs at the meridian given by 72— m, and the same numerical value with the
opposite sign occurs at the meridians given by 36— m and 36 + m. We condense into
one term the contributions of the eight points given by n, 36 — n, nt, 72— m, 36 ± m,
and take the ranges of m and n to be respectively 0 to 17 and 1 to 18. Thus, as the
multiplier of cos (mw/36) sin* (nn/36), we have an integral number which necessarily
lies between —4 and 4, and may be zero, and we have transformed the sum into a
double sum of the form

17 18 «_

_« «4 T^/ / \ flltt • j ii'Tf

2 2 F'(/i,m)cos — sin*;-,

m=0 » = I OO OO

where F' is the number in question. The most troublesome part of the process is the
determination of F'. When F' has been found it is very easy to form the sum of
such a series as that written immediately above by summing first with respect to m
and then with respect to n. When we are dealing with tessera! harmonics of the
second rank, we can thus condense into one term the contributions of 16 points of
the table, and, when the tesseral harmonic is of the third rank, those of 24 points.
Much labour is saved by going through this process, troublesome though it is, and
much greater accuracy can be secured, localise in the multiplication of cos (mtrf'SG)
by F', when F' is, say, 5 or 6, and the value of the cosine to any chosen number
of decimal places is used, it is easier to correct the figure in the last place than it is
when the same cosine occurs five or six times in a long column of figures which have
to be added together.

60. By the use of this method I computed the values of the coefficients p, &c., for
the function F (0, <j>) which is given by the -1's in the table of § 56, the 1's being
replaced by zeros. Up to the stage of summation with respect to m, inclusive, I kept
four decimal figures. Of the terms of the type

I then kept two decimal figures, formed the sums with respect to n, and multiplied
them by the corresponding numbers placed in brackets in the third column of the
table in § 58. This process gave the coefficients in the second column of the annexed
table. The integral parts only were retained. I computed the values of the
coefficients p, <fec., in the same way for the function given by the 1's in the table of
§ 56, the —1's being replaced by zero. This process gave the coefficients in the third

2 H 2

236 PROFESSOR A. E. H. LOVE ON THE

column of the annexed table. It is to be understood that in both cases the common

Q C

factors - — TT~I and (Tr/36)2 have been omitted.

F (6, <j>) = 0 or - 1.

F (9, </>) = 1 or 0.

Sum.

p

604

557

1161

1

495

329

824

r

777

630

1407

a.

350

243

593

P

295

366

661

7

-443

-223

-666

8

-291

68

-223

f

185

98

283

a

-213

-134

-347

b

- 73

- 71

-144

c

- 29

38

9

d

-338

-256

-594

e

396

351

747

/

56

26

82

ff

203

i

122

325

The Continental Blocks and Oceanic Regions as expressed by Spherical Harmonics

of the First, Second and Third Degrees.

61. I then computed the values of the harmonics expressed by (115), (116), (117),
for values of B and (f), which are multiples of 15° (or -J^TT), using first the coefficients

GRAVITATIONAL STABILITY OF THE EARTH.

237

given in the second column of the table in § 60, and then the coefficients which are
given in the third column of the same table. Finally I added the values belonging
to the same 0 and <£. The results are shown in the diagram (fig. 3), where the fine

165' ISC' 135' 120' MS' 80* 7»* 60" 46' 30* IB' 0* IS* 30* 4f 60* If XT IPS' I20* 135' I5QT

IBS' ISO* US' "20* IDS' *T W W 45* *T 15' O* 15' 30" 46' 60* 75' Vf 105' IW IJST ISO" 165'

ISO* I3y BO' K)5' 90° 75' 60* 45' Vf \V V »° 3IT 45" 6O° 7S' 9O" IPS' 120* I3S' IM' I6&-

ISO" 135- IZO* 105° 90* 7S* 60* 45* 30* IS* 0* IS1 30" 4S* 60* W 90* 105

Fig. 4.

continuous line is the contour-line along which the calculated harmonic inequality
vanishes, the heavy continuous line is the contour-line along which this inequality is
10 per cent, of its maximum below zero, and the dotted line is the contour-line along

238 PROFESSOR A. E. H, LOVE ON THE

which this inequality is 10 per cent, of its maximum above zero. It is to }>e observed
that an inequality expressed by harmonics of uneven degrees has numerically equal
values with opposite signs at antipodal points, and therefore the area on the sphere
within which such an inequality is positive is equal to the area within which it is
negative. But this equality of positive and negative areas does not hold when the
harmonics of the second degree are present. A rough calculation showed that the
zero line of the inequality illustrated in fig. 3 divides the surface of the sphere into
two unequal areas, and the inequality is negative in the larger area. The excess of
the negative area above the positive is nearly 10 per cent, of the whole surface.
The heavy line in fig. 3 corresponds more nearly than the other lines to the
principle by which geographers construct the contour-line at mean-sphere-level. The
diagram in fig. 3 suggests many features of the outline of the continental block, and
there can be no doubt that the coefficients could be adjusted so as to secure a better
agreement.* It seems best, however, to record the results as they are. For the sake
of comparison a rough map of the world is added (fig. 4). The heavy continuous line
is the outline of the continental block at mean-sphere level, and the fine continuous
line is the coast-line. I have not attempted to draw the map with minute accuracy,
and have omitted many small islands and some small enclosed patches of deep sea,
because the object aimed at is a comparison of the general features of the map of
the world with those of the diagram in fig. 3. The map is drawn by taking the
longitude east of Greenwich and the latitude of any point as the Cartesian
co-ordinates of the corresponding point of the map. Fig. 3 is drawn in the same way.
The defects of the arrangement in fig. 3, considered as representing the shape of
the continental block, are sufficiently obvious, the chief being the absence of any
indication of an Arctic ocean, and the almost complete submersion of South America.
On the other hand, the fact that even tolerable agreement in so many respects is
obtained from a spherical harmonic analysis of the extremely simple distribution
detailed in the table of § 56 may be regarded as a confirmation of the theory which
led us to assume that harmonics of the first, second, and third degrees shoiild be
predominant.

Geological Implications of the Theory.

62. The results appear to admit of a geological interpretation. We have adduced
dynamical reasons for the hypothesis that the lithosphere consolidated in a shape
which may be described as an ellipsoid with three unequal axes, with its centre
of gravity displaced from its centre of figure, and with a wrinkle upon its surface
expressed by spherical surface harmonics of the third degree ; and we have found that
the figure of the lithosphere now, as determined by difference of level above or below

* The coefficients r, t, a, b, c are especially sensitive to changes in the assumed distribution in the Arctic
and Antarctic regions where the actual distribution is least known.

GRAVITATIONAL STABILITY OF THE EARTH. 239

the geoid, is expressible, at least roughly and approximately, by means of harmonics
of the first, second, and third degrees. Now, if the shape of the lithosphere is at all
close to that in which it may be presumed to have consolidated, the inference would
seem to be that, in respect of general features, as distinguished from local irregularities,
the positions of the continental blocks and oceanic regions have not changed much
since the date of consolidation. This view has in recent times met with considerable
support among geologists.

The theory also enables us to make some attempt to indicate the general nature
of those changes which could be expected to take place. In estimating the value of
such an attempt some allowance must be made for the fact that the theory of an
elastic body in a state of initial stress is very far from complete. We try to follow
out certain clues drawn from the scanty knowledge we possess of bodies in states of
initial stress. Among these the behaviour of cast iron under tensile tests is perhaps
important. It is well known that cast iron which has not previously been tested
exhibits a stress-strain curve which is essentially different from that of mild steel, but
that, after several tests, its behaviour approaches to that of steel. It has been
conjectured that the tests have the effect of gradually removing a state of initial
stress, and thus reducing the substance to a "state of ease." That state of a rotating
gravitating planet which would correspond to a state of ease in solid bodies at its
surface would seem to be a state in which the' material would be arranged in
concentric spheroidal layers of equal density, and the external surface would be an
oblate spheroid, the ellipticity being determined by the speed of rotation and the
distribution of density ; the state of stress in the planet, when in this state of ease,
would be one of hydrostatic pressure, and the surface would be an equipotential
surface under gravity modified by the rotation. The partial reduction of the body to
the state of ease would be effected by gradual stages, prol>ably of the nature of local
fractures. Now the wrinkling of the surface, expressed by harmonics of the third
degree, arose as a consequence of the displacement of the centre of gravity and of the
ellipsoidal configuration. It would at first IKJ small in comparison with the deviations
from spherical symmetry which are expressed by harmonics of the first and second
degrees. We should therefore exj>ect that the tendency of secular change in the
shape of the lithosphere would lie to diminish the coefficients of the harmonics of the
first and second degrees. An exception must be made in the case of the coefficient e
of the zonal harmonic of the second degree ; for this coefficient represents a difference
of ellipticity of the meridians of lithosphere and geoid, and these ellipticities depend
upon the speed of rotation. When this coefficient is left out of account, the harmonic
inequality of the second degree represents ellipticity of the equator* and obliquity
of the principal planes ; the harmonic inequality of the first degree represents
displacement of the centre of gravity from the centre of figure. If the coefficients of

* G. II. DARWIN concluded from his theory of the tidal deformation of a viscous spheroid that an initial
ellipticity of the equator would tend to be obliterated. ' Phil. Trans. Roy. Soc.,' vol. 170 (1879), p. 30.

240 PROFESSOR A. E. H. LOVE ON THE

the harmonics of the first degree have ratios anywhere near to those given in the
table of § 60, the great circle along which the harmonic inequality of the first degree
vanishes has a pole somewhere in south-eastern Europe and the opposite pole in the
Pacific Ocean. The inequality is positive in Europe, most of Asia, Africa, North
America, the northern and central parts of the Atlantic Ocean, and the Arctic
regions. The effect of a gradual diminution of the coefficients of the harmonics of
the first degree would be a gradual emptying of the Pacific Ocean, accompanied
by a rise of sea-level around the shores of the Atlantic Ocean (except towards
the southern parts of Africa and South America) and around the northern and
western parts of the Indian Ocean. It has been held that such an effect has
taken place and constitutes the reason for the difference between a " Pacific coast "
and an " Atlantic coast." The ratios of the coefficients of the various harmonics
of the second degree for the two distributions considered in §§ 56-60 are widely
divergent, but they agree in leading to negative values for the harmonic inequality
of the second degree in the regions contained within oval curves which lie within
the basin of the PaciBc, and also in the antipodal regions. In a large part of the
Pacific region the harmonics of the first and second degrees reinforce each other ;
in the antipodal region they are antagonistic. Diminution of the coefficients of
the harmonics of the second degree would be manifested by a fall of sea-level in
the Pacific, and also in a region antipodal to some part of the Pacific. It may not,
perhaps, be altogether fanciful to see in the gradual reduction of area of the " Central
Mediterranean Sea " of Mesozoic and Tertiary times f the effect of a continual
reduction of those coefficients of harmonic inequalities of the second degree which
represent ellipticity of the equator and obliquity of the principal planes. Whether
these conjectures as to the particular regions which may have been affected are
acceptable or not, it can safely be said that the effects of changes in the harmonic
inequality of the first degree, and in those of the second degree which we are now
considering, would be progressive in the same sense at the same place. They would
be manifested in a tendency of the sea to fall in certain regions and to rise in certain
complementary regions and gradually to flood wide areas. The gradual character
of the positive movements of the strand-line, by which wide areas have been sub-
merged, has been emphasized by STJESS. J

The surface of the lithosphere is nearly an oblate spheroid which does not coincide
precisely with an equipotential under gravity modified by the rotation ; it is less
oblate than the geoid. The surface of a shallow ocean covering an oblate spheroidal
planet, whose outer surface is not exactly an equipotential surface, is an oblate
spheroid, and its ellipticity is a certain multiple of the ellipticity of the surface of the
planet. The ratio of the two ellipticities depends partly on the rigidity of the planet,

* E. SCESS, ' The Face of the Earth ' (Translation), vol. 2, Oxford, 1906, p. 553.
t Ibid., pp. 258, 299.
\ Ibid., p. 543.

GRAVITATIONAL STABILITY OF THE EAKTH. 241

partly on the ratio of the density of the ocean to the mean density of the planet, and
partly on the angular velocity. Owing to tidal friction, the angular velocity of the
Earth's rotation is heing gradually diminished. The effect of this is that l»th the
ellipticity of the lithosphere and that of the geoid are being diminished, and the
difference of these ellipticities is also being diminished. If, therefore, the shape of
the lithosphere were continually adjusted to the instantaneous angular velocity, the
value of the coefficient e of § 57 would diminish continually, and the adjustment
would involve a continually increasing deformation. Eventually the deformation
would be so great that the strength of the material would be too small to withstand
it, and local fractures would take place.* There is, therefore, a constant tendency
for the sea-level to rise in the polar regions and to fall in the equatorial regions, the
separation l>etween the regions of rising and falling sea-level l>eing marked by the
zero-lines of the zonal harmonic of the second degree, that is, by the parallels of
latitude, alxnit 35° N. and 35° S. This rise and fall would l>e checked at intervals by
subsidences, accompanied by series of earthquakes, in equatorial regions.

The effects produced by diminution of the displacement of the centre of gravity,
and by changes in the ellipticity of the equator and in the obliquity of the principal
planes, appear to be of a different character from the effect of diminishing angular
velocity. The former would seem to l)e spasmodic and occasional, but always in the
same sense at the same place ; the latter would appear to consist of continuous
movements in the same sense, extending over long periods, which are followed by
comparatively short periods of spasmodic change in the opposite sense.

These remarks are frankly speculative, and I am well aware that many causes
which have contributed to geological changes have l>een left out of sight. They are
put forward as tentative suggestions which, it is hoped, may prove to be of some
assistance in the solution of some of the still unsolved problems of geology.

My best thanks are due to Professor W. J. SOLLAS and Dr. H. N. OICKSON for
much kind help in regard to geological and geographical questiona

* According to a "Note" in 'Nature,' vol. 39 (1889), p. 613, this effect of diminishing speed of rotation
noted hy M. A. Bi.YTT. I have not seen the paper referred to in the " Note."

VOL. CCVII. — A. 2 I

VI. Inwutiijdtion of the IAIW of llwning of Modified ('onlite.

liij Major J. H. MANSELL, Royal Artillery.
Communicated In/ Sir A. NOBLE, F.R.S.

Received November 9, 1906, — Read February 14, 1907.

CONTENTS.

fegl

Introduction 243

Desoi'iption of apparatus 244

Characteristics of the explosive : —

1. Relation of pressure and density 246

2. The time rise of pressure, cord form 249

3. Reconciliation of the law of reduction 251

4. The time rise of pressure, tube form .251

5. The time rise of pressure, double tubular form 257

Conclusion 257

Tables (A-D) 259-262

SOME years ago the eminent French chemist VIEILLE first propounded the law of
combustion by parallel surfaces for smokeless pro|>ellaiits. By a propellant we
distinguish an explosive which explodes .from one that detonates ; and it is this
combustion by parallel surfaces which is the distinguishing characteristic of the
difference of the two phenomena.

Since VIEILLE first propounded his theory it has l)een generally accepted as correct.
Investigators, however, have not, so far as I am aware, definitely determined what
the law is. The general assumption has been that the law is of the form S = aP",
where S is the skin burnt in a given time under the average pressure P, « and ?i
l>eing constants for the given explosive.

Now, the investigators who have dealt with this subject have all done so with the
primary object of finding out what goes on inside a gun when the charge is fired.
That has also been my primary object. In fact, it is the practical as distinguished
from tin- pun-lv scientific result of the law which has appealed to all investigators.

Now, the gun is a most complex L,ras engine, and in the past has upset the most
carefully conceived and elalxjrated theories. Previous investigators have therefore

VOL. own. — A 4 IS. 2 I '2 31.5.07

244 MAJOR J. H. MANSELL: INVESTIGATION OP

gone straight to the gun and endeavoured to solve the constants a and n in the
general form of the above equation by a system of trial and error.* Some have
concluded that n is unity, others that n is ^ (!NGALLS, America), § (GossoT and
LIOUVILLE, France), 0'9 (CENTER VALL, Sweden), &c. These are wide variations, and,
as I shall show, are due in part to the form of the explosive that different investigators
have experimented with and in part to the following causes.

The principle of calculation in the gun is that the space behind the projectile is
treated as a closed vessel. Now, as the projectile moves down the bore the size of
the vessel increases. The size of the vessel therefore directly depends on the distance
the projectile will move in a given time under a given pressure. Here at once is the
difficulty, and it entirely depends upon what friction or resistance to forward movement
is assumed as to what values of a and n may be determined. This friction is made
up of (l) the resistance to engraving of the driving band, (2) the resolved part of the
rotational thrust due to the lands of the rifling, and (3) perhaps forcement of the
projectile through the gun, which is possibly conical in form at any point where the
projectile may be during its passage down the bore.

Now it is obvious that, however elaborate the theory, many large assumptions have
to be made in determining the combined effect of (1), (2), and (3), and on these
assumptions the whole resulting edifice must stand or fall.

In my investigations I tried to avoid the pit-fall of the practical application to the
gun until I was entirely satisfied that I had determined the law of burning by parallel
surfaces in a closed vessel of constant capacity. This paper, then, is a description of
the methods I have used and the results I have arrived at in my investigations. A
considerable amount of laborious arithmetical calculation has been involved, and I
desire here to express my indebtedness to Captains A. K. IZAT and C. H. NEWCOMBE,
Royal Artillery, who have rendered me valuable assistance in preparing the diagrams
and in working out some of the calculations.

Description of the Apparatus used.

Fig. 1 shows a section of the type of closed vessel used. The pressure is registered
by the compression of the copper A by the piston B. This piston carries a pen C
which traces its movement on blackened paper carried on a revolving drum D (shown
in fig. 2). E is a valve for releasing the gases from the vessel after firing. F shows
the arrangements for electrically firing the charge. The internal capacity of the
vessel I used was 28 '18 cubic inches, and its internal length and diameter were nearly
equal. The type of vessel shown in fig. 1 is unsatisfactory, because its great length
as compared with its diameter is liable to set up wave actions.

* GOSSOT and LIOUVILLE (tome XIII., ' Memorial des Potidres et Salpetres ') have recently compared
closed-vessel time rises with those calculated when using their factors. The results are not very satisfactory,
I think.

THE LAW OF BURNING OF MODIFIED CORIM I I .

•Jl,

Fig. 1. Closed-vessel apparatus.

Fig. 2 shows the end view of the apparatus and the arrangement for recording the
pressure and time.

Fig. 2. End view of apparatus.

The drum D is driven by a motor and carries blackened paper on its periphery.
G is an electrically sustained tuning-fork of the ordinary Sebert type. The one used

24G

MAJOR .T. H. MANSELL: INVESTIGATION OF

in these experiments made 500 vibrations a second. The electro-magnet H and cam
K are for the purpose of momentarily allowing the stylus on the end of the tuning-
fork to trace a record on the drum and so give its speed — the spring N pulling the
arm 0 (which carries the cam) away from the electro-magnet.

The use of the apparatus is as follows : — The charge having heen placed in the
vessel, the copper is placed in position and M is screwed home. The stylus on C is
then adjusted on the drum, as is also the stylus on G in the position of release.
The circuit of the electro-magnet H is carried by an adjustable joint to C in such a
manner that the circuit is broken by the first movement of C as the copper
compresses. The circuit being complete, the electro-magnet holds the arm O and
lifts the stylus off the drum. When the circuit is broken the arm 0 is revolved by
the spring N, and the cam K thus lowers the stylus G on to the drum and then lifts
it off again. We thus obtain a record of the speed of the drum at the actual
moment of firing. All being ready, the drum D is set in motion by the motor,
whose speed can be regulated by a rheostat. The tuning-fork is started in the usual
way, and the charge is then fired.

An example of an actual record is subjoined. These records are measured under a
micrometer with a telescopic eye-piece carrying cross wires, the telescope being
carried on a compound lathe rest. It measures centimetres to four places of decimals
in both directions of movement.

Characteristics of the Explosive.

1. If elation of Pressure and Density, — The first step in the investigation of the
burning of an explosive is to find the relation between the maximum pressure and
the density at which the explosive is fired. The explosive I have experimented with
is the latest British one, known as modified cordite. The Service abbreviation for
this is M.D. cordite, and it will be so called throughout this paper, the original type
of cordite being referred to as Mark I. In a closed vessel the maximum pressure is
independent of the temperature of the cordite, but temperature has an influence on
the time taken by the cordite to develop that pressure. The higher the temperature

IKK LAW OF BURNING OF MODIFIED COKDITK.

•247

tlu- quicker the time rise of pressure. In the gun, therefore, the temperature of the
cordite has an influence on ballistics, since at a higher temperature, the pressure
Ijeing raised more quickly, the projectile has less time in which to move forward ;
consequently there is a smaller space behind the projectile at times of equal
developments of gas, and higher pressures are therefore realised. Temperature,
therefore, is of no importance in determining the pressure-density relation, but is al'-
important in tin- investigation of time rises. I am not clear that other investigators
have borne this in mind —their publications take no note of the fact.

The pressure-density relation of M.D. cordite is shown graphically on fig. 3, and is
tabulated in Table A.

Denf.il.ie:.

Fig. 3. Pressures and densities, M.D. cordite.
The equation connecting the two variables is •

P = 360A»-54A'+698A,

where P is the pressure in tons on the square inch, and A is the gravimetric density
of loading. Since artillerists work with the Ib. as the unit of weight and the cubic
inch as the unit of volume, and since 1 Ib. of water occupies 2773 cubic inches at
60° F., density is given by the formula 2773 x weight of charge in Ibs. -r capacity in
cubic inches, and is then known as the gravimetric density.

248

MAJOR J. H. MANSELL: INVESTIGATION OF

The cubical form of this equation is of interest when one compares it with
VAN DER WAALS' general equation

^ (v-6) = KT,

V vf

which may be written

p (1-6A) = a&A3-aA2+RTA.

It would appear, then, that in any general deduction of a pressure-density relation
from VAN DER WAALS' equation omission of the term afv* is not justified. PETAVEL
neglected this term in his investigation of Mark I cordite.*

Tonkin?
2.0

19
18
17
16

15
14
13
12
II
IO
9

e

7
6
5
4
3
z
I

Cord

M.D. at A

-4

Z446

D

1265

a

e

A

o Points as measured off Che 'record .

J

calculated by Che reduction table.

: Tube

Points as
the

M.D. at A

measured
record.

calculated

0-16935"
0-07315"

allowing for excess internal
Batch. 89.

pressure

•OOI

•OO2

•OO3

•004 -005

•006

•007 -008 -009

Time- Seconds

oio

•on

•012

•013

•014

•015

Fig. 4. Time rise of pressure, cord and tube M.D.

Cordite in manufacture is made in lots, and the above pressure-density relation is
true of the general run of the cordite. In the earlier stages of the manufacture of
some experimental forms of M.D. cordite the pressure-density relation was different.
This may have been due to slightly different chemical constitution, to the presence of
an excess quantity of volatile matter, or to minor variations in manufacture. The
point is unimportant, because with experience in manufacture this variation dis-
appears; but I mention it, as in one of the experiments I shall refer to later such an

* ' Phil. Trans., A, vol. 205, pp. 357-398.

•OI6

THK LAW OF BURNING OF MODIFIED CORDITK. 249

exceptional lot was used, ami it will have to be referred to a special pressure-density
curve.

2. Investigation of the Time Rise of Pressure (Cord Form). — M.D. cordite has
heen made in various forms, some of which are only experimental. As with Mark I
cordite, the first form was cords ; since that time tubes, strips, and double tubular
forms have l>een made and experimented with.

The first time rises I investigated were, then, of M.D. cordite in the cord form. A
time rise of such a cord, measuring 0'1265 inch in diameter, fired at 80° F. at a
density of 0'2448, is shown on fig. 4. The close agreement of this beautiful curve
with the points actually measured by the apparatus used in the experiment is an
indication of the accuracy of the arrangements. Having obtained this time rise, the
next step was the investigation of the law of combustion by parallel surfaces.

The method employed was the following: — Intervals of O'OOl second were worked
to. From the curve the pressure at O'OOl -0'002, &c. second was obtained. This
pressure corresponds to a certain density of gas obtained from Table A. Now this
gas is produced by a small reduction dr in the radius r of the cord, in other words, a
skin or lamina is burnt off and converted into gas. The available capacity of the
vessel for this quantity of gas is the total capacity less the volume occupied by the
unburnt cordite. One has therefore only to solve for dr under these known
conditions. This reduction dr then takes place in O'OOl second under an average
pressure which is obtained from the time rise. The average pressures 1 have taken
are those at half time in the interval. For instance, the average pressure during the
first O'OOl second is the actual pressure shown on the curve at 0'0005 second. In
actual practice, instead of working on the reduction of radius I have worked on the
reduction of diameter.

Fig. 5 shows the results of this calculation for fig. 4 plotted in terms of reduction
in diameter and pressure. This figure also shows the lines I have selected to
represent ^the relation at temperatures of 60° F. and 80° F.

It appears quite clear that the relation is expressed by a straight line, and that
therefore the power n is unity. The equation to the lines is of the form S = aP + C,
a varying with the temperature of the cordite.

It is the existence of this constant C which has not been suspected l>efore, and
which, I think, shows the danger of assuming an equation of a theoretically perfect
form and then trying to deduce constants by trial and error.

The meaning of the constant C can only be that below about O'l ton pressure the
law of reduction in diameter does not hold. Obviously, when P = 0*, S cannot
equal C.

The cause of this change of law is, I suggest, that until some definite pressure is
attained in the vessel true explosion does not commence. I advance the following
explanation : — When the charge is first ignited, only the cordite in immediate contact
with the igniter commences to burn, Cordite being a bad conductor of heat, this

VOL. ccvu. — A. 2 K

250

MAJOR J. H. MANSELL: INVESTIGATION OP

burning does not run along the cordite rapidly. This can easily be seen by burning
cordite in the air, when it burns slowly along its length in the manner of slow-match
and the flash is not rapidly transmitted as with gunpowder. Consequently, in the
vessel the lighted ends of the cords burn non-explosively until such time as the
vessel is completely filled with flame at a high temperature. At that moment there

Ton

20

18
16

14
12
10

s>

6

4

Z

^in* / /

(

:ordit

e ac Lo'F.

,4

,'

<

+ Po

nts i

;alcul

ited

from

ii

time

&0°F.

rise

>

.'<,

/

/'

/;

/

/

/}

/

/,

y

./;

S~ f

s

"7

f,s/

/

/.

^

' +

,^

<f

/

?

-OO2 -CO! -006 -008 -OIO -012 -QM -OI6 -CHS -O2O

Reduction in diameter- Inches.

-O22 -O24 -O26 -O2&

Fig. 5. M.D. cordite — reduction in diameter in 0 • 001 second when burning under a given average pressure.

is a definite pressure in the vessel which tends to separate the cords one from another.
The cords now are lighted over their whole length and the true time of combustion
by parallel surfaces commences. The constant C is thus due to the amount of gas
produced by combustion of the ends of the sticks, when regarded as if produced by
combustion over the whole length of the stick. The error introduced into the length
of the sticks by this assumption is insignificant and can be neglected.

If this explanation is correct, one would expect the amount of cordite burnt previous
to complete ignition of the charge to be independent of the temperature. That this
is so was experimentally determined before the theoretical explanation of the constant
suggested itself to me. Undoubtedly the time to complete ignition is different with
change of temperature of the cordite, but this does not affect the ultimate time rise.
Its sole effect is a small variation in the hang-fire of the charge. Theoretically the
constant C must vary with the density of loading. It has been determined at a
density of 0'25, and the small variation at lower densities does not affect the general
accuracy of the calculation.

THE LAW OF BURNING OF MODIFIED CORDITE.

•J51

3. Reconciliation of tlie Law of Reduction. — Now the equation for reduction in
diameter in O'OOl second at 80° F. has been determined as

Redn. = 0'0013361 xP + 0'00028,

where P is the pressure in tons on the square inch.
At 60° F. the equation is

Redn. = 0'001223xP + 0'00028.

Both these results are tabulated in Tables B and C. In order to justify the
selection of the lines shown on fig. 5 as representing the above relations, I have
calculated the time rise of pressure of the charge of Cord M.D. shown in fig. 4, using
Tables A and B. The calculated points are shown on fig. 4, and below I tabulate the
results for comparison : —

Pressure in tons inch3.

Time, second.

As measured from the
record.

Aa calculated.

o-ooi

Not definitely measurable

0-17

0-002

0-4

0-32

0-003

0-7

0-57

0-004

l-l

1-01

0-005

1-65

1-61

0-006

2-37

2-47

0-007

3-4

3-56

0-008

5-05

5-08

0-009

6-95

6-98

0-01

9-35

9-3

0-011

12-1

12-07

0-012

14-95

14-95

0-013

17-5

17-57

0-014

18-95

18-99

0-014345

—

19-1

0-0144

19-15

—

The variations in pressure are within the limits of experimental error, and the
difference in total time of combustion is only 0'000055 of a second.

4. Investigation of the Time Rise of Pressure (Tube Form). — Fig. G shows the
time rise of pressure of a certain sample of Tube M.D. Cordite (Batch 88). This
class of cordite is known as M.D.T., and one of the governing factors of its rate of
burning is the thickness of its annulus. On the same figure I show the calculated
time rise of pressure of this sample, using the law I have established.

It is evident by inspection that the M.D.T. time rise does not directly follow the
cord law of reduction in diameter. The rise of pressure at the beginning is much
more rapid than when calculated as for cords. This difference presented a problem

2 K 2

252

MAJOR J. H. MANSELL: INVESTIGATION OF

full of great difficulties and which I sought to solve for a long time at the expense ot
most laborious arithmetical calculations before I arrived at the solution which I now
put forward.

Certain phenomena in connection with the burning of M.D.T. have always been
apparent and indicated the lines on which I must work. If a stick of M.D.T. be

Ton&/in? Tons/in

0— o

•
16

+

Tube M

Points &
« Cc

•• Cc
Curve ca

D. ati 80°
5 measure

ilcula.t-.ed fr
(zero

ilcula-ted aJ
internal

Iculated on
(zero noi

F. 4

j off

om a
adjus

owing
pre;

cord
/ adji-

k - -2

the

>rd n
ted).

for e

>oure.

redu
sted)

482.

recor
xiuctic

xcesa
cfcion

It

f

m
'.-

17
16

ii

1 1
O
a
u

•

9

a

7
6
5

4
9
•
I

d.
in Ca

We

II
1 1
1 1
1 1
n

i

1 1
1 1

i

17 A

tabl<

i

i

9

r

Batch i

ZS^&J

»

/ j

i
/

JPmS-o-'oattS''

' iidilir=0;0457il>"

It
1 1
1 1
i

/
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-^^

1 1
1 1
Ii
1 1

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I

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f

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g

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7,,,,. .,

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Time - .Second*.
Fig. 6. Time rise of pressure— Tube M.D. at 80° F.

ignited in the open air the burning does not proceed regularly, but is accompanied by
a succession of reports, the tube at the same time being projected about. The action,
in fact, is very similar to that of the " cracker " firework.

Now these explosions and jumps are due to the formation of gas inside the tube at

THE LAW OF BURNING OF MODIFIED CORDITE.

253

a quicker rate than it can get away at atmospheric pressure. The pressure inside
the tube therefore rises to some point at which it bursts out explosively, the tube at
the same time being projected in an opposite direction. This action goes on to such
an extent that holes are often blown through the walls of the tube.

On firing M.D.'f. in a gun, when any unconsumed is blown out I have often found
tubes with these holes or splits in their walls. The distance between these holes is
generally exceedingly regular. It is therefore clear that under certain conditions an
excess pressure exists inside the tube, even when burning under pressure, i.e.,
explosively.

Now, from the nature of my law of reduction, it follows that if an excess pressure
exists inside the tube more cordite will be burnt in a given time. There is thus
a reaction of cause and effect, and the internal excess pressure of itself tends to raise
the inside pressure more rapidly. When the gas so formed escapes into the vessel it
in turn raises the pressure existing in the vessel, and an increased rate of combustion
is the consequence.

The time rise on fig. 6 very clearly shows this acceleration of the rate of burning.
Since I saw no reason why there should be a departure from my fundamental law,
my efforts were directed to determining what the excess internal pressure was and on
what it depended. The principle adopted was as follows :—

The time rise gives the average and end pressure of any interval. The outsides of
the tubes are burning under the cord law and produce a certain quantity of
gas in the interval which can be calculated. The total amount of gas produced
is know;i from the end pressure. The difference between these two amounts
of gas has come from the inside of the tube, and hence the amount of cordite burnt
from the inside can be calculated. Referring to Table B, we determine what pressure
the inside of the tube must have been burning under to consume that amount of
cordite in the time. From this the excess internal pressure is calculated.
The results of the calculation for Batch 88 are subjoined :—

Time,

During the 0 • 0005 second interval.

second.

Average pressure

Excess pressure

Total internal pressure

in the vessel,

in the tube,

in the tube,

tons/inch1.

tons/inch2.

tons/inch1.

0-0005

0-16

4-39

4-55

0-0010

0-83

5-58

6-41

0-0015

1-45

1-95

3-40

0-0020

2-09

-1-34 (deficit)

0-75

0-0025

2-90

1-81

4-71

0-0030

:!•>:

1-28

5-15

0-0035

5-04

0-57

5-61

0-0040

6-41

-0-53 (deficit)

5-88

254

MAJOR J. H. MANSELL: INVESTIGATION OF

Now the first results of this calculation are not at all obvious. But by adjusting
the zero of the curve calculated on the cord law and comparing it with the actual
record it seemed possible that at some point, to be determined, excess internal
pressure disappeared and the tubes then burnt in strict accordance with the law in
Table B. In adjusting the calculated curve so that its general lie was in closest
agreement with the measured one, I found they crossed at about 57 tons. It then
seemed that if internal excess pressure disappears at any point it must start at some
maximum. A simple way of considering the decrease of excess internal pressure
from a maximum to zero is to consider that the internal pressure in the tube is

Ton a/in*
20

19-

tfl

Double

tubular M.O. «t 80° I

A- -2 3

17-

Poi

lt» A3

measured off

!6 the"_recor J-.

o Points ca cul&ted by
15 - table C.

Tub: M.D.

at 80°

F. A.

Batth 97

ft - 0-25037

- 0-090O3

II -

10-

7

9-

8 —

7-
6 —

.7

5-
4 —

•002

•004

•006 -008

•OIO

•014

•OI6 -OI8

Time- Seconds

•020

•022

•024

•026

•028

•030

«09» -034

Fig. 7. Time rise of pressure.

a constant until the pressure in the vessel reaches that given pressure. From that
moment internal and external pressures will be equal.

Referring to the tabulated results of total internal pressure, it will be seen that the
average of the pressures given is 4 '56 tons. This was the first pressure tried, but a
better result was given assuming 4 '8 5 tons.

The calculation, then, was made on the assumption that the internal pressure was
4-85 tons up to the moment at which the pressure in the vessel reached that figure.
The external and internal diameters of the tubes at that moment are given by the
calculation which then proceeds on the assumption that the internal and external
pressures are equal.

THK LAW OF BURNING OF MODIFIED CORDITE.

255

The points obtained by this calculation are shown on fig. 6. Their general
agreement with the measured curve is not so close as one would wish ; but for a
first attempt they appeared to support the idea that there are two phases in the
combustion of a tubular propellant : (1) when excess pressure exists inside the tube,
and (2) when this excess pressure disappears. Another batch, No. 97, was then tried.
Its time rise is shown on fig. 7. The calculation gave :—

Time,
second.

During the interval.

Average pressure
in the vessel,
tons/inch2.

Excess pressure
in the tube,
tons/inch*.

Total internal pressure
in the tube,
tons/inch1.

0-0005
0-001
0-002
0-0025
0-003

0-05
0-15
0-28
0-38
0-43

3-6
3-15
1-3

0-56
-0-26 (deficit)

3-65
3-3
1-58
0-94
0-17

Average internal P -

1-93 tons.

We now have two cases of calculated internal pressure, and the next point for
consideration was : on what does this internal pressure depend? Looking at it from a
theoretical point of view, it would seem to depend on the area of the hole and the
length of the tube. The larger the hole the more readily can the gas get away. The
longer the tube, for a given hole, the more difficult will be the escape of the gas.

In my closed-vessel experiments the cordite is cut to the internal length of the
vessel to avoid the wave pressures which occur if the cordite is banked up at one end.
The length variable does not therefore come in.

The internal radius and pressure of the two samples were : —

Batch.

97

Radius of hole.

0-04575
0-09003

Internal pressure.

4-85
1-93

Now, if internal pressure varies inversely as the area of the hole, the pressure of
Batch 97 from Batch 88 would be given by 7^ = (x.,^»») > &n&, from this, P would

be T25 tons against 1'93 tons found by the calculation. Having regard to the great
variations which may be caused in my calculations by small experimental errors, this
result was not as discouraging as it appears on the face of it.

256 MAJOR J. H. MANSELL: INVESTIGATION OF

On fig. 4 (lower curve) is shown the time rise and particulars of another batch of
M.D.T., No. 89. This batch was one of the exceptional batches I have previously
referred to, and did not show the same pressure density relation as the average run
of M.D. cordite. The pressure density curve for this batch is shown on fig. 3 (lower
curve).

Having obtained the time rise, I tested my theories by calculating a time rise under
the two-phase condition I have explained.

The internal pressure, if proportional to the inverse ratio of the area of the holes, is
1'896 tons, using Batch 88 as the standard. The points of this calculated time rise
are shown on fig. 4 (lower curve), and, except at the end of the rise, show a very close
agreement with the actual condition of affairs.

The end of the rise shows disagreement. But if the actual rise of Batch 89 be
compared with the others, it will be seen that the falling away of Batch 89 is a most
exceptional condition of affairs for M.D.T. Whether the falling away was due to
experimental errors or to some chance peculiarity of an exceptional sample I was not
able to determine, as there was no more of the batch left.

The determination of the length influence on internal pressure requires a closed
vessel of different dimensions, and I have not dealt with this aspect.

From the visible behaviour of M.D.T. when burning in air it is obvious that special
actions are taking place. I venture to think that the calculations and experiments
I here set forth support the theory that in the combustion of tubular propellants
there are two distinct phases : the first when excess pressure exists inside the tube,
and the second when internal excess pressure has ceased. With such a complicated
problem it is clear that those investigators who have only had tubular forms of
propellants to deal with would be faced with a most intractable problem in
endeavouring to discover the true law of combustion by parallel surfaces. It is this
difficulty which in part accounts for the various formulas which have been advanced.

Another somewhat important consideration is that, if you assume an equation of
the form S = aP" for a tubular propellaut, all tubes that have the same annulus
should give the same ballistics. It seems clear from experimental firings in guns that
the size of hole for a given annulus has an influence on ballistics. There is no
explanation of this fact in the simple equation formula, but it is at once explained by
the system of calculation which I have here set forth. The system also explains the
splits in the tubes and all the various phenomena connected with the combustion of
tubular propellants.

At the same time it is possible to obtain also for tubes a reduction equation of the
form Redn. = aP + C. I originally obtained an equation of this form, which is set out
in Table D. By this table I am able to calculate time rises very approximately with
various tubes. But it is liable to break down, gives no explanation of the various
phenomena, and is scientifically unsatisfactory in that there is no reason why the
fundamental law of reduction should differ for tubes from cords.

THE LAW OF BURNING OF MODIFIED CORDITK. 257

Whilst admitting that Table D is merely an expedient, it otters the advantage of
la-ing quicker to work with, and the results are fairly consistent with the tubes as
supplied.

5. Investiyrttiim of the Time Rise of Double Tubular M.D. Cordite. — It will be
clear from a slight consideration that with the propellant in the cord form a
decreasing surface is exposed as combustion proceeda With the tube form an
approximately constant surface is exposed. If it were not for excess internal
pressure the surface would actually be constant. For if dr is the skin burnt
at any time, and R and r the external and internal radii, the original surface
is proportional to 2ir(R+r) ; so when a skin dr is burnt the surface is proportional to

Without going deeply into the science of internal ballistics it will be apparent that
the longer maximum pressure can be sustained in the gun the greater will be the
efficiency of that gun for a given length as regards muzzle velocity. Of course, there
are limiting conditions as "regards the capability of the gun to withstand this
sustained maximum pressure, but such considerations are outside the scope of this
paper. Speaking generally, however, there may be advantages in sustaining the
maximum pressure in a gun. Now a double tubular form will present an increasing
surface as combustion proceeds, and this will tend to sustain the maximum pressure in
the gun.

The time rise of pressure of a sample of a double tul>e is shown on fig. 7 (left-
hand curve). The dimensions of this double tul>e are given on the figure, the firm
lines showing the actual shape which in manufacture had not come out as true arcs of
circles. A mean circle was therefore determined for the purpose of calculation, and
the adjusted double tube is shown by the dotted lines.

With the view of showing the results given by Table D, I have calculated the
time rise of the double tube, using that table. The calculated points are shown on
the figure. It will Ije seen that there is a very close agreement in the curves, except
at the beginning. The error at the beginning is, of course, due to the excess internal
pressure effect being greater with the double than with the single tube.

The American powder is a multitubular one, that is, short cylinders pierced with a
number of holes. Excess internal pressure would have a very magnified effect on
such a powder, and this, I think, accounts for the wide difference in value of the
exponent of P as used by INUALLS.

Conclusion.

In the foregoing I ha,ve, after giving the reconciliation of my law for the cord form,
confined myself to the cases where it apparently fails. I have endeavoured to show
the cause of the failure and at the same time present the solution. I have not given
examples of time rises of the strip form, for with them there is no disturbing cause so
long as manufacture is not varied.

VOL. CO VII. — A. 2 L

'258 MAJOR J. H. MANSELL: INVESTIGATION OF

Undoubtedly it would be a great convenience in working if the integration of these
curves were possible. Much thought has been given by different investigators in the
past to this problem and much mathematical ingenuity has been displayed. But in
these problems one does not obtain expressions which are directly integrable, and
assumptions and approximations have necessarily to be made. Such approximations
give exceedingly good results within limits, but when one comes to their application
to the gun, and its many variables, the limits are so widened that a break-down under
certain conditions is an ever-present danger.

I have, therefore, preferred to follow the system adopted by Mr. BASHFORTH in his
calculations of extended trajectories, that is to say, I break up my time-rise curves
into small arcs, and, assuming a mean pressure for the interval, find from the
calculated end pressure if my assumption has been correct. If not, I have now a
guide to the mean pressure to assume, and so on. In this manner each arc can
generally be calculated in three trials, and with practice many arcs are obtained at
the first attempt.

The application of this law to the practical case of the gun is outside the scope of
this paper, and it is obviously undesirable to publish such investigations in connection
with English ordnance.

For reasons which I have alluded to, the application to the gun presented more
complications than the experiments which I have here outlined.

Having adopted certain frictional laws for the gun, based on the law of burning
which I now put forward, I have found that the application holds over a very wide
range of varying conditions of loading and calibre, when using cords which is the
form with which we have most experience. There can be no higher test than this of
the fundamental truth of the law.

THE LAW OF BURNING OF MODIFIED CORDITE.

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[ '263 ]

VII. On t/ie Dispersion in Artificial Double Refraction.

By L. N. G. FILON, Af.A., D.Sc., Fellow and Lecturer in Mathematics of University

College, London.

Communicated by Professor F. T. TROUTOK, F.R.S.
Received January 25, — Read February 28, 1907.

SUMMARY OK CONTENTS.

Page

§ 1. Introduction 264

PART I.
THEORY OF THE EXPERIMENT AND DISCUSSION OF THE VARIOUS ERRORS.

§ 2. Simple theory of the experiment 265

§ 3. Description of the apparatus - 268

§ 4. Effect of introducing the polarizing Nicol 27:?

§ 5. Effect of finite breadth of the source 274

§ 6. Effect of relative rise and fall of the two beams and of elastic yielding of the bed-plate and

knife-edges 274

§ 7. Influence of obliquity on relative retardation 275

§ 8. Combined effect of flexure and obliquity - .... 276

§ 9. Imperfect adjustment of the inclination of the slit . . 277

$ 10. Imperfect horizontal adjustment of the knife-edges 277

$11. Imperfect vertical adjustment of the knife-edges 278

§ 12. Error due to weight of beams themselves 27R

§ 13. Error due to imperfect annealing 27H

PART II.

EXPERIMENTAL RESULTS.

§ 14. Glasses observed ^>7;»

§ 15. Linear law connecting A. and the stress 280

§16. Significance of this linear law 284

§ 17. Methods of reduction 287

§18. Tables of results 289

§ 19. Discussion of the values of AO, &WU 29*

§ 20. Systematic residuals 293

§21. Possible explanation by absorption bands 297

§ 22. Determination of absolute values of C 301

§ 23. Effect of chemical composition on stress-optical properties 302 .

§ 24. Failure of HOOKE'S law for glass 2783 303

§ 25. Conclusion 305

VOL. CCVII. — A 419. 26.7.07

2C>4 DR. L. N. G. FILON ON THE

§ 1. Introduction.

IT is well known that glass and other transparent isotropic substances, when compressed
unequally in different directions, behave like doubly-refracting substances and exhibit
the colours of polarized light. Attention was first called to this by FRESNEL
(' Annales de Chimie et de Physique,' vol. XX.), and by Sir DAVID BREWSTER
(' Phil. Trans.,' 1816). For further investigations in this field, reference may be made
to F. E. NEUMANN (' Abhandlungen der k. Acad. v. Wissenschaften zu Berlin,'
1841, II. ; see also ' Pogg. Ann.,' vol. LIV.) ; to CLERK MAXWELL (' Trans. Roy. Soc.
Edin.,' vol. XX., Part I. ; or ' Collected Papers,' vol. I.) ; to G. WERTHEIM (' Annales
de Chimie et de Physique,' ser. 3, vol. XL., p. 156); to J. KERR ('Phil. Mag.,'
October, 1888) ; and to F. POCKELS (" Uber die Anderung des optischen Verbal tens
verschiedener Glaser durch elastische Deformation," 'Ann. d. Physik,' 1902, ser. IV.,
vol. VII., p. 745). Of these only WERTHEIM and POCKELS have considered how the
effect varies with the nature of the light employed.

If homogeneous parallel light is passed perpendicularly through a plate of thickness r
which is subjected to principal stresses P, Q in its plane, these stresses being uniform
throughout, then it is found that the light on traversing the plate is broken up into
two rays polarized in the directions of principal stress. The relative retardation in
centimetres of these rays on emergence is given by

E = (fr-p^T,

where fii, /*a are the indices of refraction of the two rays.

Now experiments have shown that fii—p.3 is very approximately proportional to the
principal stress difference in the wave-front, P— Q. Whether this is true for high
values of P— Q is not certain, and some experiments to be described in the following
pages (see § 19) will show that the proportionality of pi—^ to P— Q in all cases must
still be regarded as doubtful. Assuming, however, this law, which is certainly very
nearly true in most cases, at all events when P, Q are stresses of the same type
(tensions, or pressures), we have

R = C(P-Q)r,

where C is a coefficient depending only on the nature of the material and on the
wave-length of 'the light used. This coefficient C will be spoken of in what follows
as the "stress-optical coefficient."

WERTHEIM, from observations of a uniformly compressed block of glass through
which he passed successively (i.) sodium light, (ii.) white light, (iii.) white light filtered
through a red glass, stated the following law :—

The relative retardation in air is constant for all colours. In other words, the
stress-optical coefficient C is independent of the wave-length ; the difference of the
refractive indices is therefore likewise independent of the wave-length, that is, the
double refraction due to elastic strain exhibits 'no dispersion.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 265

S, in his more recent investigation, observed the artificial double refraction
in a number of Jena glasses. His observations, though not primarily intended to test
the effect of colour, nevertheless gave exceedingly valuable results in this connection,
insomuch as POCKELS experimented with three different kinds of nearly homogeneous
light, namely, those of sodium, lithium, and thallium. The results would therefore
be far more precise than those obtained with very mixed colours by WERTHEIM. They
show that, in certain glasses, the stress-optical coefficient does vary with the wave-
length, being numerically greater in the green than in the red ; and in very heavy
lead glasses this variation is more rapid as we approach the blue end of the spectrum.

Some years ago the present author, being at the time unaware of POCK ELS'
experiments, devised a method of observing the variation of the coefficient C
continuously throughout the spectrum, the object being to test the exactness of
WKRTHEIM'S law.

An account of this method, which was modified and improved from time to time,
and of the experiments undertaken to carry it out, will be found in the
•Camb. Phil. Soc. Proc.,' vol. XL, Part VI. ; vol. XII., Part I. ; vol. XII., Part V.
These experiments amply confirmed the results of POCKELS. They also showed that
the chief desideratum for obtaining accurate results was that the stress in the glass
slab through which the light was passed should be sufficiently uniform. Now the
compression apparatus which was used by the author, and by previous experimenters,
suffered from the defect that it was practically impossible to adjust it so as to obtain
a uniform pressure in the slab of glass under observation. Moreover, what adjustment
could be made was long and difficult, and could be attained only by trial ; it apjieared
further that this adjustment was disturbed, in a way that could not be calculated and
allowed for, when the load was altered. This greatly reduced the accuracy expected.

An apparatus was then devised, with a view to obtaining a system in which the
stresses should l>e known exactly and in which the optical effects should IK- the same
as those due to uniform pressure in a slab of constant thickness. For the purposes of
this research a Government Grant was kindly placed at the disjHwal of the author by
the Royal Society, whereby the necessary apparatus could be constructed and the
expensive glasses required for the research purchased. The present paj>er is an
account of the experiments carried out with the new apparatus and of the results
reached.

PART I.

THEORY OF THE EXPERIMENT AND DISCUSSION OF THE VARIOUS ERRORS.
§ 2. Simple Thewy of the Experiment.

l-« t N, F (fig. 1) be two rectangular slabs of glass, whose cross-sections are shown
in the figure. The slalw are bent in a vertical plane by couples without shear, whose
axes are horiaontd and parallel to the plane of. the cross-section. How such couples
are obtained will be explained subsequently.

VOL. OCVII. — A. 2 M

266

DR. L. N. G. FILON ON THE

The horizontal and vertical sides of the cross-sections of N and F are (2a1; 26,),
(2oj, 2b3,) respectively, and the centres of the two cross-sections are O1 and O3.
Let S be a point-source of light ; S' its image after passing through N ; PI, P3 the
points in which any ray through S meets the vertical midplanes of N, F respectively.

Fig. 1.

Let P\ be the image of P1; after a single refraction at the inner face of N (the
one towards F), and P'3 the image of P3 after a single refraction at the inner face of
F (the one towards N).

Then it is evident that S', P\, P'2 must be in one straight line.

h = height of S above a fixed horizontal plane,
2: = „ Q! „ the same plane,
*a » '-'a » >» »

2/i = ,, PI » Ox,

y» = „ P» » O3,

d = distance of S from the nearer face of N,
I = „ between midplanes of N, F,
P-i) Ma = refractive indices of N, F respectively.

We have

SS' = 204 (/*,
and since S', P/, P,' are collinear

or, writing
(1) becomes

= a, *i-

P3P'a =

(2),
(3),

Suppose now monochromatic light proceeds from S.

Let GI, C3 be the stress-optical coefficients of the two slabs for this kind of light,
M], M3 their bending moments reckoned positive when the slabs are bent concave
downwards.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 267

The relative retardation after traversing the first slab for any ray which passes
through P, at a small angle to the horizontal = 2a, . C| (3Miyi/4a1/>1J).

For, although the stress is not uniform along the path of the ray, the mean stress
along the ray = stress at the middle point, since the stress varies linearly as the
distance from the neutral axis. Also the length of the ray differs from the breadth
of the slab only by quantities of the second order. Hence the result above.

Accordingly the total relative retardation, after passing through the two slabs, is

given by

...,., (4).

Substituting for yt from (3) into (4),

R = (3M./26,3) (^.-(SM^fe,') C^+(3M,/2V)Cs[(2l+yI) (i + 0-)-crfc] (5).
Now, if R is to be independent of y,, we must have

= 0 ....... (6).

This condition will not of course be accurately fulfilled for all colours at the same
time ; in the first place, because C, and Ca do not in general follow the same law of
variation for the two slabs ; in the second place, because <r contains ft, and therefore
involves the wave-length.

It is easy to see that the latter cause of error is quite negligible. For if d be
large compared with a, or a,, which was the case in all the experiments, the error
introduced in o- by a variation 8/x in the refractive index (taking /i, = /*,, which is
practically true) is approximately

Now, a, + a, = 3 centims. in the experiments to be described ; d = 250 centims.
about (or larger), 8p. = O'Ol between the C and F lines of the spectrum which
represent fairly well the extreme range of the observations.

Hence, ft being about 1 '5,

(oi + o^S/i/pU = -01/187-5 < -00006.

Accordingly the error introduced by this cause would correspond to an error in C,
of less than 6 in a hundred thousand, an error which is absolutely negligible, since
the errors of ordinary observations in the method to be described amount to ffaj or
T&o of C. (See ' Camb. Phil. Soc. Proc.,' vol. XII., p. 58.)

The di fii-rent variation of d and C, with the wave-length would be far more
important This, however, need not be considered, for the two slabs N and F are
taken from the same cast, so far as possible, so that C, and C, should be identical.
In some cases it was found that C, and C2 differed ; but, at the same time, the

2 M 2

268 DR. L. N. G. FILON ON THE

experiments showed conclusively that for slabs of the same material Cj and Ca
remained very approximately proportional one to the other for all the values of X
examined. In this way condition (6) is satisfied independently of the wave-length.
It follows from (5), using (6), that the relative retardation is given by

(7).

Accordingly the relative retardation, after the light from such a point-source has
traversed the two slabs, is the same for all the rays from S. Two such slabs are
therefore optically equivalent to a single slab which would be under an accurately
uniform tension.

By properly adjusting the differences of height, zl— z2, z1—h, the amount of relative
retardation may be adjusted within certain limits.

In general, I will be chosen small with respect to d. Thus a- is a proper fraction,
say of order y^. Equation (6) thus shows that M] and M2 are to be chosen of
opposite signs, and approximately equal in magnitude.

This gives at once the physical explanation of the result (6). The rays pass
through approximately horizontally. If we compare two rays passing through at
different levels, the ray which passes through the regions of greater tension in N
passes through the regions of lesser tension (or greater pressure) in F, and the two
variations balance one another.

Further, since the amount of relative retardation as given by (7) involves only the
relative heights of the axes of the slabs and the source of light, the latter may be
moved parallel to the axes of the slabs without affecting the relative retardation.
Hence a horizontal line-source, parallel to the axes of the slabs, may be used instead
of a point-source. This was, in fact, indispensable in order to obtain the required
intensity.

§ 3. Description of the Apparatus,

Light from an arc lamp L was passed through a condensing lens C and through a
thin horizontal slit T (fig. 2), which was placed from 1\ to 3 metres away from the

Fig. 2.

glasses and straining apparatus. It was polarized by a Nicol P, whose polarizing
plane was roughly at 45° to the horizontal, and then passed through the two slabs N
and F. These were adjusted so that their levels differed very nearly by \ centim.

lUSl'KKSION IN ARTIFICIAL DOUBLE U.FIIACTION. 269

The two slabs were cut from the same piece of glass, and every precaution was taken
to ensure that they should )>e as nearly identical us possible. The dimensions of the
cross-sections were practically the same, namely, in the notation of the last section,
2<»i = 2at = 3 centims. and 2&i = "2bt = 1 centim.

The length of each slab was about 13 ceutims.

Bending moments of opposite sign, in a vertical plane, were applied to these, so
that the light passed through parts of the glass either altogether under tension, or
altogether under pressure, according to the manner in which the bending moment
was applied. Of the method of applying such bending moment a fuller account will
be given below.

After emerging from the two beams the pencil of light traversed a Nicol A, which
was crossed with the Nicol P. It was then focussed by a cylindrical lens Y (which
consisted in practice of a glass beaker filled with water) upon the vertical slit of a
spectroscope Q and the spectrum observed in the usual way.

The condensing lens C was focussed approximately upon the Nicol P ; both 0 and
Y were introduced in order to improve the illumination. It was found otherwise
that so much light was lost that only a very faint spectrum could be obtained, and
this was useless for the purpose of the observations.

The latter consisted in measuring accurately the position in the spectrum of the
black bands corresponding to light completely quenched between the Nicols P and A.
Light of any colour will, of course, be quenched between crossed Nicols when the
relative retardation of the two rays (polarized in horizontal and vertical planes
respectively) in which it is split up by the strained glass amounts to an integral
multiple of the wave-length.

Referring to formula (7) this occurs when

-A)] ....... (8),

71 being an integer. If C, were independent of the wave-length, as WERTHEIM'S law
would require, then, for a baud of a given order, n is fixed and the wave-length X of
extinction is proportional to M»

If, however, C, varies with X, then X/C, is proportional to M*

By observing the values of X corresponding to a given Mf, and varying M,, we
obtain the relative magnitude of the coefficient C, for these varying values of X.

The probable error of setting on the centre of a black band was calculated by the
author in the ' Camb. Phil. Soc. Proc.,' vol. XII., Pt. V., pp. 314-315, and was found
to be about 1', so that the wave-length of extinction is determined with a propor-
tional error smaller than 0-002. The average error due to inaccuracy in setting the
cross-wires in the eye-piece upon the centre of the black baud is then about 8 to
10 tenth-metres, so that the wave-lengths may be considered known accurately to
three figures.

The bending moments were applied to the slabs by means of the apparatus shown

270

DR. L. N. G. FILON ON THE

in fig. 3. The slab G rested on two knife-edges R and S. On it rested two other
knife-edges U and V, supporting a graduated steel bar L Fixed to the top of I was
a triangular knife-edge K, from the projecting extremities of which hung two

«---"--* a- ...*.-€. ^

S

*

N

r^K'

Iluul

,,,,,„,,!

mi hi.i

lln|1 t(|1J

a

7j

\

{

\

j r

i>

c

(f

1

_j

V\
R

01

B

[ .T"

P ^

T

1

A

I

T

j

i i

I

'fl

o

i

i
L

;

W

Fig. 3.

symmetrical hangers A. These passed through holes cut in the bed-plate P, which
supported the whole apparatus, and by means of a cross-piece C and another hanger
H a load W was applied which acted on I vertically downwards at its middle
point.

In order to ensure that the reactions between G and the knife-edges B, V should
be vertical, the knife-edge R rested on steel bicycle balls B, so that it would readily
move under horizontal friction ; V was made a double knife-edge, the plane containing
the two edges being carefully adjusted to be vertical. The knife-edge U could slide
along I and be clamped in any desired position. When U was clamped and the load
applied, the apparatus was perfectly stable, the knife-edge S being kept in its place
by the friction of the bed-plate. In order to prevent the knife-edges from cutting
into the glass and breaking it under the large loads applied, four small slips of steel Q
were inserted between the knife-edges and the glass. These distributed the actual
stress without altering the actual statical resultants, and at points near the centre of

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 271

the slab the effect of such local perturbations must be negligible. (See ' PhiL Trans.,'
A, voL 201, pp. 114, 145.)

When the apparatus is in perfect adjustment K is exactly midway (measured
horizontally) between U and V, and the horizontal distances between the edges of V
and R and of U and S are equal. If these be each o, then the bending moment
applied to the part of the slab between R and S is constant and equal to £cW.

For, since the reaction at the upper edge of V is vertical and the load at K is
vertical, then the reaction at the lower edge of U is shown to be vertical by
considering U, I, K as one system. Thus the reactions at the lower edge of U and V
are each equal to £W. Again, the reaction at the upper edge of R is vertical and
therefore the reaction at the upper edge of S is also vertical. Hence these reactions
also are equal to £W.

Also it is to be noted that, if the adjustment be perfect, the bending moment
applied to the beam or slab is a pure bending moment. There is no total shear across
any cross-section between R and S.

In such a case it is well known that the distribution of stress obeys accurately the
Euler-Bernouilli laws and consists only of a tension My/A/,J parallel to the axis of the
beam, where M = applied bending moment, y = height above neutral axis (horizontal
line drawn through the centroid of the cross-section in the plane of the cross-section),
A = area of cross-section, k = its radius of gyration about neutral axis. The formulae
(4) and (7) are therefore verified.

In fig. 3 the knife-edges U and V are outside R and S. The bending moment is
therefore positive, with the convention of p. 266. For the second beam the arrange-
ment is the same, except that now U and V are inside R and S, so that the bending
moment is negative.

The difference of height between the slabs was obtained by placing the knife-edges
R, S for one of the beams upon two steel blocks of height 0'5 centim. instead of
directly upon the bed-plate. The bed-plate itself was a solid plate of steel, very
strong and resting upon two heavy tables T of the same height.

In the alwve description no account has been taken of a large number of small
errors which must theoretically affect the method.

The principal are the following: — (1) In the theory explained in § 2 modifications
will be introduced owing to the fact that a polarizing Nicol is introduced in the path
of the rays of light lietween the source and the slabs. (2) The source of light is not
a line-source, but a slit of finite breadth. (3) When the load is applied, the middle
part of one beam rises and the other sinks: thus the heights zt, z, and the relative
height 2,— 2, in formula (7) are not fixed. (4) The bed-plate P and the tables T are
nut absolutely rigid. This will alter 2, and 2,, but not 21—2* (5) The rays do not go
through the glass horizontally and at right angles to the axes of the slabs, and the
assumption that the mean retardation = retardation at mid- point of path is only an
approximation. (6) The slit used as a source of light is not accurately horizontal.

272 DR. L. N. G. FILON ON THE

(7) The knife-edges are never quite accurately adjusted. (8) The weight of the
beams themselves will affect the stresses. (9) The beams are not always perfectly
annealed and the permanent stresses in the glass modify the appearances.

In the following sections the corrections due to these errors will be investigated.

§ 4. Effect of Introducing the Polarizing Nicol.

We shall now consider the effect of introducing the polarizing Nicol upon the
inclination of the rays of light. In order to estimate the magnitude of this effect, it
will be sufficient to treat the Nicol as a singly-refracting substance. If the larger
index of refraction be adopted this should, in general, give us an upper limit to the
error introduced. If no sensible disturbance is found to be thus introduced, we may
assume that this will be the case in the actual experiment.

Let S (fig. 4) be the source of light, P the image of a point of the mid-plane of the

Fig. 4.

nearer slab, viewed by refraction through the face nearest S. If the Nicol were not
present the light would travel along the line SP. In consequence of the introduction
of the Nicol it travels along the broken path SCDP.

Let <j>, i/f be the angles of incidence and refraction, x the angle which SP makes
with the normal to the faces of the Nicol.

Let the perpendicular distances of S and P from the nearer faces of the Nicol be m
and n, and the thickness of the Nicol be t.

Then

(m + u) tan <j> + t tan ^ = (m + n + t) tan x,

or, writing t((m+n + t) = y,

tanx— tan (f> = y (tan «/»— tan <£) (9).

Using p. sin «/i = sin <f>,

tan t/» = tan <f> [y + (/x2- 1) tan2 c^]'1 a.
Hence (9) becomes, retaining only first powers of y,

tan x- tan <£ = y tan x ([/**+ (ft2- l)tanax]~1/a-l) .... (10).

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 273

Now in the experiments y < T^ an(l the iimximum variation in \ for rays passing
through the slabs amounts only to 5Jfl.
But from (10)

A(tanx-tan^) = ysec»x{/iV+^t-l]tan>x)-^-l}. . . (11).
a\

Accordingly the greatest variation in tan X— tan <f> for the rays passing through
the slahs is less than

)-*--!} ..... (12).

To compute the order of this expression take p. = 1'5 and x = 30°, which last is an
extreme estimate. (12) gives

8 (tan x- tan <£) = -0-000033,
that is,

S(x-<£) = -0-000033 cos* x = -0-000025 about.

The change in relative level of the points P,, P, of fig. 1 due to the above amounts
to (0-000025) I, and in the experiments I — 18 centims. roughly. Thus the change in
relative level = 0'00045 centim. The proportional change in the total effective
stress = change of relative level -f- semi-height of slab = 0'0009, and this will
produce a negligible error in the stress-optical coefficient. Accordingly for mono-
chromatic light the effect is to increase the obliquity of all rays by a small constant
amount, or to change the effective height k of the slit. As the absolute value of h is
not known, and will be found not to enter into the calculations, the presence ot the
Nicol will not affect the observations for monochromatic light.

Kin- white light, however, it may do so if the quantity tan <j>— tan x vary sensibly
with the colour of the light used, other things remaining the same.

From (10)

8 (tan x~ tan <f>) = — /ny8/i sin x (/*'— 8in* x)~*"«

Now for calcite

X = 6708, jt. = 1-484, /x, = 1'653,
X = 5350, p.. = 1-488, p.0 = T658,

8/x. = 0-004, 8/io = 0-005.

Taking the ordinary index as the basis of computation and x = 30° as before,

8 (tan x- tan <f>) = -0'000026 nearly.

The proportional error in C deduced from this is obtained by multiplying by
/^» t-e., by 27. It is therefore 0-00070. This error corresponds only to the
VOL. ocvn. — A. 2 N

274 DR. L. N. G. FILON ON THK

dispersion between the lithium and thallium lines. The error will be greater when
we reach the violet end of the spectrum, but it will still be too small to affect the
observations.

In experiments demanding great accuracy it might be desirable to polarize the
light before it passes through the slit. The accuracy possible under the present
circumstances did not seem to justify this additional complication in the apparatus.

§ 5. Effect of Finite Breadth of the Source.

It may be shown that if the slit have a finite breadth '2e, the intensity of the light
that gets through is proportional to

sin q0

0

where

0 = 27rK0/A, g = o-e

HO being the relative retardation corresponding to the middle of the slit.
In the actual case g = J-^Q approximately.
The minima are given approximately by

0 = 2nrr (!-&•).

The proportional error in the wave-length of extinction is therefore BTSOO- which
is negligible.

Also the minimum no longer corresponds theoretically to perfect darkness, but with
a slit between ^ millim. and 1 millim. wide the bands were very dark and quite
definite.

§ 6. Effect of Relative Rise and Fall of the Two Beams and of Elastic Yielding of

the Bed-plate and Knife-edges.

Owing to the elasticity of the glass, the middle parts of the beams will undergo a
vertical shift owing to flexure, and the bed-plate and apparatus as a whole will sink.
In consequence we have variations 8z,, Sz2, §h depending on the applied load.
Thus the right-hand side of equation (8) is multiplied by a factor

This may be allowed for by supposing M2 (or W) multiplied by the same factor,
equation (8) remaining otherwise unaltered.

The effect is then to add to the applied weight a correction

W[8zI-8z2+o-(8zl-8/t)]/[z1-22+o-(zl-A)].

Now the relative rise and fall of the beams themselves is an elastic effect and may
be taken, in such a small correction, strictly proportional to the load.

DISPERSION IN ARTIFICIAL DOT'KLK RKFRACTION. 275

The linking of the lx'<l-|>]at<> was measured experimentally and found to be elastic
in its nature, the recovery being eomplete.

Generally the experiments showed no trace of permanent set, the readings being
the same when unloading as when loading.

We may safely assume therefore that 82, — 8z3+<r (8:, — 8/t) is pmjxirtional to W, so
that the correction to be applied to W ou account of these errors is of the form

KW».

The value of K is uncertain and depends very largely on the circumstances of each
experiment.

Using KVKUKTT'S and AMAOAT'S values of YOUNG'S modulus for glass (i.e., between
600,000 and 700,000 kilogs.-weight per square centimetre), the part of K due to relative
rise and fall of the two beams was calculated to be about 0'0004. Thus for
W = 50 kilogs. the proportional correction is as high as 2 per cent.

The part of K due to the sinking of the l>ed-plate was found experimentally to be
of order 0'00026. Also the experiments could be arranged in such a manner that the
two corrections operated in different senses; and this precaution was always taken.
Their combined effect will give K of order O'OOOl, and even for the highest loads used
the correction will be small.

In practice this correction KWa was determined from the observations themselves,
in a manner explained in § 17. For most sets of observations it was found to be
insensible.

£ 7. Itijhn'nn- of Obliquity on, IMntivc- Retardation,

We may consider the glass .as optically made up of a series of horizontal homogeneous
layers. In passing from one of these layers to another, the refraction takes place
approximately in a plane perpendicular to the optic axis.

1 1 will l>e sufficient for our purpose to consider a ray passing through in a cross-
section, that is, in a plane throughout perpendicular to the optic axis. If the
curvature of such a ray be negligible, we may take it that we can neglect the
curvature of all oblique rays.

Now if n be the index of refraction at a point in the glass distant y from the
neutral plane,

/^,, Ix-iiig tin- index of refraction for the unstrained glass. It may then be proved that
the curvature of a ray passing through nearly horizontallv is approximately /<///.

Now it has been shown by KKRR ('Phil. Mag.,' October, 1888), and by POCKKLS
('Ann. d. Pliysik. P.MIJ. p. 745), that the absolute variation due to stress in the
index of irtraetimi for either ray is of the same order as the difference in the two
indices due to the same cause. In general, for the highest stress employed, the

2 N 2

276 DR L. N. G. FILON ON THE

(/!„— juor)-gradient is of the order 10~4. Taking ^0 = f , the curvature is of order :-j 1 0~*.
Hence the greatest deviation from the straight line = £ (curvature) (thickness of
slab)' = 3 x 10~* — a divergence which cannot possibly affect the results.

Thus we are justified in treating the paths of the rays as linear. Moreover, the
divergence of the ordinary and extraordinary rays after refraction at entrance
= fji~a (fjL^— fior) (angle of incidence) very nearly, and it is easily verified that the effect
of this is also entirely negligible. Therefore we may treat the two rays as geometrically
coincident.

The paths of the rays being linear, the planes of polarization are fixed throughout.
For these can be proved to be the plane through the ray, and the line of strain and
the plane through the ray perpendicular to the first plane. And the line of strain is
always parallel to the axis of the slab.

Also if ft = angle between ray and line of strain, the relative retardation introduced
by an element ds of path is

Hence the total relative retardation is

R = 2aCT0siria£secr,
where

T0 = tension at mid-point of path,

2a = thickness of slab,
y = inclination of ray to the horizontal perpendicular to the axis.

In practice the limits for cos ft are + 0'01, and for y are ±0'02. It follows that the
factor sin3 ft sec y introduces a proportional error less than 10~4 in the relative
retardation. It may therefore be altogether neglected.

§ 8. Combined Effect of Flexure and Obliquity.

The relative vertical displacement of the two slabs due to flexure varies with the
cross-section taken. Now the pencil of rays used passes through a comparatively
large region of the beams, extending to about 1 centim. on either side of the central
cross-section. It is readily shown that the changes in zlt z2, due to flexure as we
pass from the central cross-section to sections distant x from the central one are
given by

82, = - Sc.o'W/ielWi* = -3W/E]

Mf x=\,
VV = +3W/EJ

the slab N being bent concave downwards and F concave upwards.

The greatest possible change in Szl— Sz2, due to this cause, is numerically equal to
<;\\r/E or (taking E = 600,000 kilogs.- weight per square centimetre) = Wx 10~6.

The proportional correction in the stress amounts to 10~5 W/^— 2.,) nearly, i.e., to

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 277

10~6.2W. Thus for the extreme load of 50 kilogs. it is only 10"*, and may be
disregarded.

§ 9. Imperfect Adjustment of the Inclination of the Slit.

If the slit he not horizontal its different parts will act as different sources of light
at different heights h.

It is clear that if the inclination be too great the different parts of the slit will
give different dark bands in the spectrum, all overlapping. The integral band will te
diffuse in consequence and not readily measureable. It is quite easy, however, to
make this adjustment to a nicety, as follows :—

Let AB (fig. 5) be the slit, A'B' the image of AB in the cylindrical lens (Y of fig. 2)
for rays proceeding in a horizontal plane. Then each element P of the slit gives a
vertical line of light through P'. Let 8,8.,
be the opening of the slit of the spectro-
scope. The latter is a good deal smaller
than the image A'B', so that in practice
only a moderate length of the luminous slit
is used. If now the cylindrical lens be

moved to one side or the other, so that Fig. 5.

S,S, travels from one end of A'B' to the

other, then, if the luminous slit be not horizontal, the band will shirt in the spectrum
in consequence. When no such shift occurs, we know that the adjustment is very
exact. There is very little difficulty in making this adjustment, and accordingly
there is no reason for anticipating any sensible error from this cause.

§ 10. Imperfect Horizontal Adjustment of the Knife-edges.

In practice it is impossible to ensure that the two pairs of knife-edges shall be
exactly symmetrical with regard to the vertical through the load. Failure to satisfy
this condition introduces shearing stresses in the l)eams, so that the axes of
polarization are no longer horizontal and vertical and further the bending moment
varies from cross-section to cross-section.

The complete analytical investigation of the correction in this case is long and
difficult, but the results may he summed up as follows, for the simplest case, when
only one of the slabs is supposed imperfectly adjusted.

In general there is no longer perfect extinction, so that the band is not quite black.

Assuming that the " overlap " of the two slabs is half their height, the position of
the band for rays passing through the edge of either slab is unaltered.

The position of the band for a ray passing exactly at mid-level is shifted towards
the red end of the spectrum by an amount not exceeding O'G of a tenth-metre.

Thus, remembering that rays which have passed through the glass at different

278 DR. L. N. G. FILON ON THE

levels correspond to parts of the spectrum also at different levels, we see that the
band is no longer straight and vertical, but curved, the convexity being towards the
red. This convexity is, however, so small that it would not in any case be observable.
If the condition (6) of § 2 is not exactly satisfied the band will still be straight,
to a first approximation, but no longer vertical. Thus when the bending moment
varies from cross-section to cross-section for light passing to the right of the
mid-section the band is tilted one way, for light passing to the left it is tilted the
reverse way. The integral effect will be that the thickness of the band will no
longer be uniform, but the band is still symmetrical with regard to a vertical line,
corresponding to light going through the mid-section. The settings which are made
on the middle of the baud are therefore unaffected.

§ 11. Imperfect Vertical Adjustment of the Knife-edges.

It will also happen that the knife-edges will not all be at exactly the same height,
so that the axes of the two slabs are not exactly horizontal and parallel. The effect
will be that for rays passing through in a plane distant x from the central section

Zi— z2+a-(zl — h) = A + Bx

instead of being exactly constant, B being a small coefficient.

This will broaden the band and render it more diffuse, but will not shift its centre.
Observation shows that this effect must be very small, as, in general, the band is very
well defined.

§ 12. Error due to Weight of Beams themselves.

In computing the stresses no heed has, so far, been paid to the fact that the
weights of the glasses themselves will introduce certain stresses in the slabs. The
weight of each slab is on the average 120 grammes. This, although very small
compared with the total load in most cases, may introduce a small error in the case
of the band of the first order, which corresponds to a smaller load.

For the beam N the weight of the glass was found to introduce practically no
bending moment in the centre, as the supports were very nearly at the quarter and
three-quarter span points.

For the beam F the moment introduced is the same as if the weight on this slab
were increased by exactly its own weight.

It is quite easy in practice to eliminate this by adding a small counterpoise to the
weight on N.

§ 13. Error due to Imperfect Annealing.

We now come to the only error — with the exception of that due to rise and fall—
which is sufficiently important to be allowed for in the reduction of the observations.

IMSI'I.KSIOX IN AUTIFICIAl. HMPBLE REFRACTION 27U

The annealing of the glasses used, which wen- supplied by Messrs. /KISS of Jena,
was found to be by no means perfect. In some cases this was revealed even by
a cursory inspection between crossed Nicola. In other cases, the glasses being
unloaded, a one-wave plate of selenite was introduced between the Nicols, its axes
being horizontal and vertical. This showed a black band, on the same principle that
the strained glass shows such a band.

Now if the glasses had MO residual stress the relative retardations should \te the
same when the azimuth of the axes of polarization of the selenite plate is altered
by 90°.

If there 1« residual stress, however, it will affect the light differently in these two
rases and tlif I mud will lie si lilted. In iin.sl cases tin- existence ••!' MI<-!> a n -si dual tfcnOt
was exhibited very plainly by this method. As a rule the l>and due to the selenite
plate was straight and vertical, showing that the residual stress was fairly constant.

If AT, AU, AS be the three components of residual stress in a vertical plane
parallel to the axis of the lx;am, then the axes of polarization make an angle <f> with
the horizontal, where

tan 2<£ = 2AS/(T + AT-AU)

and the principal stress difference

P-Q = v/(T + AT-AU)3+4(AS)3.

If we neglect squares of

AT/T, AU/T, AS/T

it is easy to calculate that the retardation

2C,T,a, +20,7,0,

has to be increased by

20, (AT,a1-AU1aI) + 2Cs (AT^-AU/t,),
or, taking

c, = c, = c

in these corrective terms, the retardation must be increased by

20 [ATiOI-AUlo,+ AT/ij-AU/iJ,
and this is equivalent to putting in a constant correction W0 to W.

PAKT II.

EXPERIMENTAL RESULTS.
§ 14. Glasses Observed.

The glasses used in this research were made for me by the firm of Zmss in
Jena. The makers bein^ unable to communicate to me the chemical composition

280

DR. L. N. G. FILON ON THE

of the glasses, the latter were analysed for me by Mr. W. J. REES, on the staff of
Messrs. CHANCE BEOS. To Mr. REES' skill I am indebted for the following results :—

Numl>er of glass . .

1809.

3453.

2783.

3296.

935.

3413.

3749.

SiOo

l>er cent.

35-4

IMT cent.

68-1

per cent.
52-7

per cent.

67-5

]>er cent.

32-5

per cent.

31-6

per cent.
70-2

PhO

18-7

—

31-6

—

28-2

23-6

—

ALA,

3-7

—

0-6

—

8-5

8-0

—

ZnO

—

—

1-2

—

—

—

—

MgO

0-5

5-4

—

> 0-4

—

—

—

B-A

34-3

5-7

1-4

15-4

27-7

33-0

5-9

K20

7-4

20-8

12-5

16-7

3-1

3-8

23-9

The majority of these glasses belong to the borosilicate variety, excepting 2783.
2783 is a flint glass, and was stated by the makers to be identical in composition
with another glass, O 154, the composition of which (see ' Camb. Phil. Soc. Proc.,'
vol. XH., Part V., p. 314) was stated by Messrs. ZEISS to include Na2O and BaO.
It seems probable that the composition of the later glass is a little different to that
of O 154.

§ 15. Linear Law connecting X and the Stress.

Since it was known beforehand that corrections to W of the type W0 + KW2 would
have to be applied, W0 being due to the imperfect annealing, and KW2 to relative
and absolute rise and fall (see §§ 6, 13), instead of calculating the stress-optical
coefficient C directly, as was done in previous experiments, the relation between W
and X was first studied, with a view to disengaging the corrections.

In practice, readings for W and X were taken for both first and second orders
of the band, and even, where possible, for third orders — for both tension and pressure.
The tension and pressure observations were obtained by altering the relative heights
of the two slabs by interchanging two steel slips which raised the supports of one
of the slabs. The bending moments were not altered.

A typical set of results is embodied in Table I. below.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION.

281

TABLE L— Observations of Glass 1809.

A

B.

C.

w,.

X.U.

AX.*.

SAX.*.

wt.

x*.

AX.*.

SAX.*.

w*

x,^

AX^

2AXOU.

14-25

4506

26-25

4460

38-25

4480

15-25

4891

385

28-25

4832

372

40-25

4705

225

16-25

E906

315

30-25

5167

335

42-25

4909

204

429

17-25

5570

364

1064

32-25

5494

327

1034

44-25

5151

242

446

18-25

5808

325

1004

34-25

5820

326

988

46-25

5375

224

466

19-26

6265

370

1059

36-25

6180

360

1013

20-25

6600

335

1090

38-25

6490

310

996

A'

F.

C'.

W_,.

X*.

AX.^

SAX^

W_2.

A*.

AX,^

3AX^

W_,.

X0^

AX^

SAX.*.

11-25

4430

24-25

4460

38-25

4570

12-25

4750

320

26-25

4773

313

40-25

4815

245

13-25

5000

250

28-25

5040

267

42-25

4975

160

14-25

5310

310

880

30-25

5330

290

870

44-25

5172

197

602

15-25

5600

290

850

32-25

5618

288

845

46-25

5360

188

545

16-25

5890

290

890

34-25

5905

287

865

48-25

5560

200

585

17-25

6160

270

850

36-25

6215

310

885

60-25

6760

190

578

18-25

8460

300

860

38-25

6500

285

882

VOL. CCVII. — A.

2 o

282 DR L. N. G. FILON ON THE

The parts A, B, C refer to observations for tension : A', B', C' to observations for
pressure. In the columns headed Wa (n = ±1,2, 3) are placed the observed weights,
the suffix n indicating the order of the band observed, bands in pressure observations
being taken of a negative order. The same notation will be kept throughout. The
columns headed X^ contain the observed value of the wave-length of the light quenched,
in tenth-metres. They are deduced from circle readings of the spectroscope, the law
connecting these circle readings with the wave-lengths being obtained from
observations of a known comparison spectrum. The spectrum of the arc between
carbons soaked in calcium salt was used for this purpose.

Now, if we look at Table I., A., under the heading AX,,,.,., we see that the differences
of the observed X for unit differences of W have a fairly constant average value, as is
well shown on taking differences corresponding to differences of W of three units.
This is done in the column headed 3AXob8 of Table L, A.

It would seem, therefore, that the relation between X and W is approximately
linear. This impression is found to be confirmed when differences are taken in
Table I., B, C, A', B', C'. In each case the differences are sensibly constant,
especially if we bear in mind that an experimental error of 1 0 tenth-metres is to be
expected.

There are some local inequalities, some of which will be shown later to be probably
significant, but as a first approximation it seems we may assume a linear relation
between X and W.

Fig. 6 shows the observed X plotted to W for the set of observations of Table I., B'.
The observations obviously lie very close to the straight line given by the equation

X = 949 + 145W.
This equation was obtained by assuming a formula

X = Xo + fcW .......... (13).

k = -—= was obtained by taking the mean value of the differences in the columns

(JL W

headed 3AX,)t<i of Table L, B', and X,, was then determined from the condition that
the best straight line must pass through the centre of gravity of the observations.
The equation (13) clearly leads to the relation

or, R being the relative retardation for a band of «th order,

R = n*W/(l-Xo/X) = C»Tr/(l-X,,/X),

where T = effective tension, T = thickness, C0 = a constant independent of the
wave-length.

DISPERSION IN ARTIFICIAL 1MMT.I.K RKl-'i: ACTION.

289

Tims, the stretss-optical coefficient C (= /*,— /ia for unit stress) is given by the

approximate formula

C=C0/(l-A0/X) .......... (14),

or

(15)

The curve connecting 0 and X is therefore a rectangular hyperMa. When X
C = oo , and as X increases without limit, C decreases to its limiting value CQ.

6500

6000

5500

5000

§4500

4000

25 30

LOAD IN KILOS

35

O

Best fitting line, X = 949+ 145W.

Observations.

Line, X = IT

Fig. 6. Typical diagram showing relation of X to W.

Thus, if the law continued to hold accurately for small wave-lengths, then for light
of the critical wave-length X,, the stress-optical effect would become actually infinite.

No ildiilit this law is purely empirical so far, and will very probably not hold for
very small wave-lengths. It is, however, sufficient to indicate that, as we approach

2 o 2

284 DR. L. N. G. FILON ON THE

critical values in the ultra-violet, the stress -optical effect will very likely be largely
increased.

In order to show the accuracy with which the observations above determine the
value of X0, the straight line passing through the origin and through the centre ot
gravity of the observations has also been plotted on fig. 6.

Its equation is

X = 175-35. W,

and on looking at the diagram it is obvious that no such straight line can fit the
observations.

Further, it will be shown that all the observations, not merely of the glass 1809,
but of the six other glasses examined, conform to a first approximation to the
linear law.

§ 16. Significance of this Linear Law.

We have now to enquire how far this linear law has a physical meaning otherwise
than as the expression of the trivial result that within a certain range of values all
continuous variation is approximately represented by a straight line.

In previous papers, when C and X were the quantities plotted, the relation was not
well expressed by a straight line, the observations lying, in some cases, on a very
decided curve (see ' Camb. Phil. Soc. Proc.,' vol. XII., Part V.). The observations
which led to such curves were therefore re-reduced. The glasses selected which
showed the effect most strongly were the Jena lead glasses O 1 52 and S 57.

Both of these were very closely fitted by a linear relation between X and W.
How close the fit was may be inferred from Table II. below, which shows the
observed and calculated values of X for one set of experiments with S 57, which is a
very heavy flint glass, containing 80 per cent, of PbO.

In the table, W denotes pressure in kilogrammes applied by means of a compressing
apparatus described in the paper referred to, and the entries in the column headed
X,.., are computed from the formula

X = 3 1 24 '9 + 4 -51 36 W.

The other sets of observations of S 57 and O 152, which have been re-reduced,
show equally good agreement between the observed values of X and those calculated
from a formula of type (13).

Now the mean residual in Table II. is less than 5 tenth-metres, whereas the
probable error of determination of the centre of a band is about 10 tenth-metres.
Thus the law appears to fit the observations as closely as is possible within the limits
of experimental error. It is worth noting that with these glasses, which contain a
high percentage of lead, no deviations from the law, such as will be shown later to
take place in some borosilicates, appear to exist.

Nevertheless one important experimental fact throws doubt on the universal
validity of the linear law, even for lead glasses. POCKELS has shown (' Ann. d.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION.

285

TABLE II. — Observations of S 57 (re-reduced).

w.

A*

A^

A^-X^.

A-.-W

W.

A*.

A^

AOW-A^

A^-A,^

318-4

4550

I.V.-J

-12

-49

547-8

5599

5591

- 5

+ 9

343-4

4677

4675

+ 2

-23

572-9

5701

5711

-10

+ 6

364-0

4771

4768

+ 3

-13

695-0

5819

5810

+ 9

+ 23

389-2

4878

4882

4

-12

620-2

MM

5924

1

+ 9

409-8

4985

4975

+ 10

+ 10

639-5

6016

6011

+ 5

+ 12

438-0

5090

5088

+ 2

+ 7

664 7

6124

6125

1

+ 2

455-7

5178

5182

4

+ 6

685-7

6216

6220

4

- 8

480-8

5800

5295

+ 5

+ 20

710-8

6341

6333

+ 8

3

501-5

5387

5388

1

+ 13

735-6

6447

6445

+ 2

-19

526-6

5600

5502

- 2

+ 15

—

—

—

—

—

Physik,' 1902, p. 745) that for a glass containing l>etween 60 and 70 per cent, of
Pb( ) the stress-optical coefficient changes sign, and an experiment made by him with
such a glass pointed to the fact that the stress- optical coefficient did not vanish
simultaneously for all colours, a result which has been independently confirmed by
the present author from considerations of curves showing C and AC'/ AX plotted to
percentage of lead (see ' Camb. Phil. Soc. Proc.,' vol. XXI., p. 335). Now if the law
C = C0/(l— X,,/X) held universally, the vanishing of Cu would imply the vanishing of C
for every wave-length.

Moreover, it seems impossible to find theoretical justification for such a formula.
It is well known (see DRUDE'S ' Theory of Optics,' cap. V., pp. 388, 389)* that the
index of refraction p. is given by the formula p.1 = l + SA^/jl — (X^/X)*}, where
X,, = wave-length in. racuo of light belonging to one of the natural periods of the glass.

[*Nolt atkM July 3rd, 1907. — Throughout the paper I have followed DRUDK. But if we adopt
LORENTZ'S formula, viz. : —

or similar formula1, the most essential part of the reasoning remains in most cases practically unaltered.]

286 DR. L. N. G. FILON ON THK

It seems, at first sight, highly probable that the effect of stress will be, not to
introduce different free periods of the atoms for differently polarized rays — that is,
not to alter Xp — but to change the coefficient Ap, which depends on the number and
arrangement of the molecules.

This will lead to the result

or

Now n itself, when expanded in powers of 1/X, will involve terms in X~a, X~*, etc.
Therefore C will involve only such even terms. Hence no formula involving X to odd
powers can be theoretically acceptable.

If we suppose that one term, corresponding to wave-length X,,, is active in
producing the dispersion, both in ordinary refraction and artificial double-refraction,

we have

......... (17),

X/) ........ (18).

The formula (17) is open to the same objection as C = C0/(l— Xo/X), namely, that it
does not satisfy the case of a glass where the double -refraction vanishes for one
wave-length without the dispersion vanishing at the same time. It is clear that in
this case other free periods, whose effect is usually negligible, become important.

For other glasses, however, the formulae (17) and (18) might be good approxi-
mations. To get fj. from (18) remember that for wave-lengths greater than 4300 the
dispersion terms are <^ of the whole. Then, using the Binomial Theorem, we find
that, to an accuracy of YbVo nearly,

Hence

C = Cp/[A;Vo{l-(yx)2}] = Cy{l-(X'p/X)>} ..... (19),
where

'

A formula of the type (19) for C would lead to a curve connecting the wave-length
of extinction and the load of the type

In general, when \'p and X,, are small, it will be found that either formula,

c = c((/{i-(x,,/x)}, c =

represents the observations almost equally well.

I'ISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 287

But in the case of the results of Table II. a hyperbola was fitted to the observations,

its equation being

X-(3460)'/X = (6-2489) W.

The differences between X as calculated from the above formula and X observed nre
given in Table II. under the heading X(A..— Xhyp. The mean value of the residuals
taken without regard to sign is between 13 and 14, or nearly three times the value
of the mean residual from the straight line.

Even, this mean residual hardly exceeds the probable error of observation, so that
this would not be conclusive against the hyperbola. But an examination of the
individual numbers in the last column of Table II. shows strong systematic positive
residuals in the middle and negative residuals at the ends, and these systematic
divergences certainly suggest that the hyperbola is not the most suitable curve.

The index of refraction of this particular glass is tolerably represented for the
visible spectrum by the formula

-39125/{l-(2159-6/XY'}.
Thus

X, = 2159-6, /t'0 = 1-5107, A', = 0'39125.

From this X',, of formula (20) comes out to be 19247. This differs entirely from the
value obtained from the experiments, namely 3460. We are thus led to the
interesting conclusion that in this glass at least the free periods which produce the
ordinary dispersion are probably not active in producing the dispersion of artificial
double refraction.

This removes theoretical justification in this case for the formula

c = cy{i-(xyx)'},

even if it had not been shown inferior as a purely empirical fit.
We may then provisionally accept the law

C = C0/(1-X0/X),

and the results in what follows will be reduced with reference to it.

At the same time it must be remembered that the physically significant formula
is prol>ably of type (16). It will be shown in § 21 that even in the visible spectrum
there are local divergences from the linear law.

§ 17. Methods of Reduction.

In lilting :i linear law

X = A. + i-'W

to a set of observations, the corrections due to the sinking and permauent stress had

ti> In- taken inti>

288 DR. L. N. G. FILON ON THE

The correct value for the load is given by

It is therefore a formula

X = X0+F(W0+W+yW2)
which has to be fitted.

Assuming for the present that for monochromatic light the relative retardation is
strictly proportional to the load, we have, for the band of the rth order, kf = k/r, where
k is the same constant for observations of all orders. Thus

rX = -rAo + fcWo + i-W + fcyW'.

Suppose we have p observations, the first step is to take a number of differences AX
corresponding to differences AW sufficiently large to minimise the effects of accidental
irregularities and to form the fraction

2(AX)/2(AW) = {(AX)/(AW)}r,

the suffix r denoting that the band observed is of the rfh order.

If our p observations correspond to values of W differing by a constant increment
AW, and if we take differences of X corresponding to differences <?AW, we obtain
p—q equations—

If these be added up, we have

r\ . 1
iq(p-

-X. -AW =

. .
q(p-q) .=1

where W = mean value of W.
We have then

........ (21).

A comparison of the values of r (AX/A W)r then enables us at once to discover whether
a correction yW3 is needed for the observations or not.

For most of the glasses examined the values of r(AX/AW)r do not indicate a
correction of this type of sensible amount. In doing the reductions for such glasses y
has been taken equal to zero.

For one glass y has a sensible value. In this case a suitable value of y having been

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 289

formed, corrections were applied to the observed load and the observations re-reduced
with y = 0.

From this point onward, therefore, y may be taken zero in the reductions, k is
then equal to r (AX/A\V)r.

In practice the values of r(AX/AW)r vary slightly with different r's. In most
cases, li.i \\r\rr. a sitHiricnt ly I^MM! tit is nlitaineil l>y taking for /• tlir iiu-an \alin- of
r (AX/AW)r and reducing the observations of different orders by means of this single
value. In one glass MIMIKK'S law did not seem to hold quite exactly, and the

observations of different orders were reduced independently.

|
k having been determined, AO+ - W0 is found from the condition that the best fitting

straight line

X = Xo+-W0+-W. (22)

r r

must be satisfied by the mean values X = X, W = W.
We thus obtain equations

A,=

2A, =

3A3 = 3Xo + JtW0 = 3X3-

etc.

Two of these equations are theoretically sufficient to determine Xo and W0. In
practice three are often obtained. The three equations are then solved by least
squares. The solution is given by

Xo = (3A3-A,)/2,
kW0 = (Ai + 2A,+3As)/3-2Xo.

From the values of k, Xo, W0 so determined X has been computed from the
formula (22) and compared with the observed value.

§ 18. Tables of Results.

The following gives a table of the constants Xo, /fcW0, k for the various sets of
observations. Observations corresponding to tension and pressure are distinguished
by the letters T, P respectively.

When X,,, k, kWt are known, the wave-length of the band of r*h order is computed
from the load by the formula (22). The average discrepancy in tenth-metres between
X thus calculated and X observed, for each set of observations, is entered in the column
headed (O-C).

VOL. CO VII. — A. 2 P

290

DK L. N. G. FILON ON THE

TABLE III.

Glass.

A*

JfcWo.

k.

0-C.

1809 T

352

-670

340-25

15

1809 P

770

407

389-07

11

3296 T

436

83

323-89

14

3296 P

638

-76

316-58

20

3453 T

. 439

113

249-61

18

3453 P

609

-89

246-52

17

3413 T

419

20

311-68

14

3413 P

687

-69

299-24

6

3749 T

405

194

258-48

12

3749 P

724

-92

249-04

18

935 T

183

-1320

376-19

27

935 P

719

929

252-21

30

For the glass 935 a correction yWs was applied to W, y being taken +0'001 for
pressure and — O'OOl for tension. In (22) we have then to substitute W-f yWa for W.
This glass is badly annealed and does not seem well fitted by the formula.

The glass 2783 had to be reduced differently. This is a lead glass, a specimen of
which had been examined under simple pressure and whose behaviour had appeared
peculiar (see ' Camb. Phil. Soc. Proc.,' vol. XII., Part V., p. 323, where the glass in
question is described as O 154).

There are two sets of tension observations, denoted by A, B in Table IV., and two
sets of pressure observations denoted by C, D. The values of Ar, ?-AX/AWr (see § 17)
are given in Table IV.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION.

2'J1

TABLE IV.

Set

r.

A.

rAA/AW,,

0-C.

A

1

386

266-46

18

2

361

262-30

8

B

1

607

265-64

18

2

443

262-80

11

C

1

968

236-70

26

2

687

245-96

17

3

608

252-00

16

D

1

768

243-58

14

2

675

247-36

16

In this case there seems to be a progressive increase of 7-AX/AW,. with the order, for
both tension and pressure. This excludes a correction for sinking, since the latter
must act opposite ways for tension and pressure. It is here probably due to a failure
of HOOKE'S law, which the observations have shown otherwise, and which will be
discussed in a later section.

Under the circumstances no real advantage could be derived here by attempting
to reduce the various sets by means of a single formula. The sets have, therefore,
been independently reduced, using the formula

X = A, + W(AX/AWr).

The actual observations of all glasses are given in Table V. for purposes of reference.
Rich column corresponds to a single set of observations. As a rule the order of the
band observed will be clear from the place of the observation in the series. Wherever
this is not so, or where observations of different orders correspond to the same load,
Roman numerals have been added to indicate the order of the band.

2 P 2

292

DR. L. N. G. FILON ON THE

TABLE V.

•

W. 321

M.

34

>3.

34

13.

37

19.

93

5.

!

.•-.':.

w.

11-25

4430

4549

11-75

4321

12-26

4750 4519

4485

4817

13-25

5000 4826

4791

4810

4598

4983

14-25

4508

6310 5083

6064

4888

4875

5230

15-25

4891

5600 5449

5388

4312

4249

5182

5182

4558

4375

4380

4553

5517

4271

4357

4646

4507

16-25

5206

5890 5788

5706

4652

4543

6485

5468

4800

4812

4680

4705

4881

5759

4581

4715

4824

4750

17-25

5570

6160 6085

6027

4874

4793

5812

5789

5063

4900

5280

8036

4830

4924

5082

4938

18-25

5895

6460 6444

6379

5056

6000

6119

8079

5325

5312

5160

5200

5633

8375

5041

5143

5250

5213

19-25

6265

5343

5273

6418

6383

5557

5425

5987

6596

5323

5438

5494

5444

20-25

6800

5818

5508

5846

6840

5875

5710

6367

5572

5689

5759

5698

21-25

5855

5747

8085

6089

5900

8663

5858

6938

5963

5938

22-25

6105

5987

6340

6340

6165

6130

6106

6197

6231

6190

23-25

4232

8402

6241

6565

6450

6345

6452

6483

6425

24-25

4460 4401

4409

6813

6549

6880

4298

6569

8687

6838

8703

28-26

4460

4773 4746

4780

4524

4678

4274

4628

28-25

4832

5040 5038

5045

4836

4881

4655

4874

29-25

4278

4285

4300

30-25

5167

6330 5378

5388

4270

4264

5111

5185

4377

4374

4495

4430

5024

5100

4295

4412

4411

4429

32-25

6494

5818 5701

5898

4647

4570

5449

5484

4685

4687

4680

4700

5376

5380

4600

4680

4687

4671

34-25

5820

5905 8020

6000

4804

4818

5786

5771

4924

4928

4930

4935

5717

5670

4841

4928

4904

4908

35-25

4240 III.

36-25

6180

6215 6372 II.

6340 II.

5013

5020

6092

6071

5189

5201

5180

5185

6076

5942

5107

5220

5113

5167

36-25

4381 III.

4488 III.

38-26

6490 II.

6500 II. 8850 II.

6830 II.

5270

5281

641811.

6391 II.

5424

5432

5425

6445

6410 II.

6208 II.

5372

6489

5374

5394

38-25

4480 III.

4570 III.

4379 III.

4480 III.

4321 III.

4327 III.

39-25

4721 III.

4811 III.

4483 III.

4437 III.

40-25

5508

5505

6680 II.

5706

5712

6880

5895

6698 II.

8514 II.

5832

5750

5635

5836

40-25

4706 III.

4815 III.

4615 III.

4676 III.

42*25

4909

4975 5017

6059

5768

5768

4834

4879

5957

5998

5930

5945

4827

4756

6900

6003

5885

5888

44-25

6151

5172

5996

8008

6024

5077

6218

6241

6175

8220

6157

6267

6135 II.

6125

44-25

4299 III.

I :.-_•:,

5344

6388

5184

4993

mi

6376

5360

6283

6274

6234

6281

6464

8483

8465

6418

6527

6410 II.

6387

"• •

4501 III.

i- •

: 6871

5889

6507

6518

5430

6473

6485

5283

8683

6760

..." 1L

6703

48-25

4880 III.

60-25

6750 |

6636

6676

5715

4843

61-25

9003

6010

5840

un

I'.'.ti

.M •

6163

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 293

§ 19. Discussion of the Values of X0, &

The first thing which strikes the eye on looking through the results of the last
section is, that although tension observations of different orders and pressure obser-
vations of different orders are fairly well fitted by the same AO, Wu, and k, the same
does not hold of tension and pressure ol>servation8 taken together.

The differences in k are only what should have been expected, since k depends on
the adjustments.

With regard to kWa the values for pressure and tension should theoretically be
equal and opposite. For if light traverse a thickness r of glass in which a residual
tension Tn exists, a term CT,,T is added to the relative retardation when external
tension is applied and subtracted from it when external pressure is applied.

Again A,, should l>e a constant for the glass, and therefore the same for tension and
pressure, if the stress-optical coefficient l>e independent of the nature and magnitude
of the load applied.

Now Table III. shows clearly that, although the values of kWn differ in sign, they
are only very roughly of the same order of magnitude.

Possibly this might be accounted for by the fact that in different experiments the
light did not pass through the same parts of the glass, so that the value of the
residual stress miglit have been different.

The divergences in X« are considerable ; AU appears systematically larger for
pressure than for tension.*

§ 20. Systematic Residuals.

The residuals A,,,,, — \nl have in all cases been plotted on a large scale against Al4l.
Three of these diagrams are shown in figs. 7 to 9. The pressure and tension olwerva-
tions have l)een plotted to a different base in each case, to avoid the diagrams over-
lapping, so that two zero-marks appear on each scale of residuals. Of these the
upper zero mark refers to tension ol>servation8.

Most of them (e.g., fig. 9) are fairly irregular, which is not surprising when we bear
in mind that an error of 1 division in the ordinate (10 tenth-metres) is the probable
error of the observations.

Two glasses, however, 3296 and 3453, figs. 7 and 8, appear to show very strongly
systematic residuals between 4200 and 5500. If we look at fig. 7 we notice that the
curves rise from 4200 to a peak about 4700, after which there is a sharp fall with a
trough alxnit 5050.

AtttT this the curves run fairly horizontal, with indications of another peak at 6300.

[* Note added April 3r<1, 1907. — Later experiments do not confirm the systematic difference between the
v.-ilurs of A,, fur tension ;m<l pressure. Very prolwbly the divergence previously noted was due to a change
in the adjustments which had to be made when passing from tension to pressure, and which rendered the
observations of the two kinds not strictly comparable.

The true value of A.,, appears to be the mean of the values obtained for tension and pressure.]

294

DR. L. N. G. FILON ON THE

The course of the diagrams in fig. 8 is very similar. There is a well marked peak
about 4700, followed by a depression in the neighbourhood of 5000. There are also
slight indications of a depression at 6000, with a subsequent rise. •

4200 4500 5000

WAVE-LCNCTHS IN rf/vm-Affrsrfs.

5500

1st order obe.

-- Ilnd
— Illrd

6000

% Tension obs.
© Pressure „

6500 6700

Fig. 7. Glasses 3296. Diagram of residuals.

These systematic residuals, which are in most cases quite large, and which are
shown in the same place by all the tension and pressure observations of these glasses,
cannot be chance effects. Neither can they be affected, denoting, as they do, compara-
tively rapid changes in \, by any of the slowly varying corrections which have been
discussed. They can be accounted for only in the following ways : —

(1) Possible erroneous identification of a spectrum reference line in the neighbour-

hood and consequent wrong determination of \ observed ;

(2) Bad division errors of the spectroscope circle ;

(3) Bad error in some of the weights employed at these points ;

(4) Systematic change of personality of observer in this neighbourhood, due to

change of colour ;

(5) Actual variation of the law of stress-optical effect in this neighbourhood.

(1) is ruled out by the fact that in the glass 3453, where the effect was first

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION.

295

noticed, it was discovers 1. not from any such curve, but from the actual circle
readings of the spectroscope, which usually increased by steadily increasing differ-
ences ; in this glass, just after the readings corresponding to X 5000, the difference

decreased instead of increasing. Keai li ll^s Were taken several times \\itli '_'le:it care.

and the effect was confirmed in each case. This demonstrated that the cause was
not accidental.

4500 5000

//V T{A/TH-M£TR£S.

1st order olw.

Ilnd

5500

6000

• Tension obs.
© Pressure „

6500 6700

Fig. 8. Glasses 3453. Diagram of residuals.

Clearly a wrong determination of a reference line is out of the question ; this could
not cause an irregularity in the differences of circle reading.

With regard to (2) and (3), the weights and the divisions of the circle were tested
with great care and found correct.

As to personality, the jump of 40 tenth-metres between X 5000 and X 4800 would
require an error of 6' in locating the centre of the band, and a change of personality
of this amount, in a fairly bright region of the spectrum, is unthinkable. Besides, if
the effect is due to such a change, it should appear in all the glasses, which is not the

c ise.

Under these conditions it oeoma safe to assert that between wave-lengths 4500 and

296

DR. L. N. G. FILON ON THE

5500 there exists a definite deviation from the straight line law, which deviation
takes the shape of an undulation, with a crest at about 4700 and a trough l)et\veen
5000 and 5100.

To trace this effect more exactly, a new set of readings were taken with the glasses
3453. The readings were taken with special care at intervals of load of half a kilogramme.
The residuals from the best straight line were computed as before, and they are

4?00

4500

5000 5500

T£MTH-M£Tft£S

1st order obs.

6000

6500 6700

- Und

- Illrd

» »»
Jl >»

• Tension obs.
© Pressure „

Fig. 9. Glasses 3413. Diagram of residuals.

shown on fig. 10. Here, the observations being much more numerous, the residuals
indicate a distinct curve. The dotted curve in the figure has been drawn freehand
through the points to give some idea of the general shape of this curve.

It is seen that this curve amply confirms the previous set of results, although the
observations were taken at several months' interval, with different adjustments, and
probably different personal equations.

On examination, only 3296 shows anything like so marked an effect. 1809 and
2783 show the effect in much the same place, but weakly, and some sets of
observations do not confirm it. 935 and 3749 are hopelessly irregular. Nothing
definite can be asserted about them. As to 3413, the pressure observations do not
show this effect at all, and the tension observations show it only very doubtfully.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 297

There arc indications, however, in this glass, of a systematic dip at 5500, and a
subsequent rise. Various glasses also show signs of a peak in the red, between
6200 and 6500. None of these, however, are more than mere indications, and it is
only the curves for 3453, and in particular fig. 10, on which any safe deductions and
measurements can be based.

If we refer to the table of § 14 we see that 3453 and 3296 are very much alike in
chemical composition. Apart from this, no relation between this effect and chemical
composition can be predicated. It seems almost certain that boric acid has nothing
to do with it. The glasses richest in BjO3 do not show the effect. K,O can hardly
be the explanation, or 3749 should show the effect more strongly. It seems not at
all unlikely that a small impurity, such as magnesium or zinc oxide, may be the
cause of the result. It is noteworthy that the only glasses which seem to show the
effect at all definitely, are precisely those which contain MgO and ZnO, and that the
one which really shows the effect in a measurable manner contains quite a respectable
percentage of MgO.

§ 21. Possible Explanation by Absorption Bands.

The shape of the curve of residuals resembles the curve of index of refraction
plotted to either period or wave-length when we pass through an absorption band.
This suggests that the effect may be due to some faint or latent absorption band of
the glass in the visible spectrum, which band corresponds to a period active in
producing the artificial double-refraction.

Following, as before, DRUDE ('Theory of Optics,' cap. V.), we have, ft being the
index of refraction, and K the co-efficient of absorption,

p.3 (1— ix)3 = terms not depending upon the absorption band

1- , + Q/{1-M«/X-(W} ...... ..- (23),

where X,, is the wave-length of the absorption band, a is a coefficient which increases
with the absorption, and Q is a coefficient depending on the arrangement and number
of the electrons.
This leads to

{i-

Now suppose the stress T to leave X, and a unaltered, and to alter Q.

r = 2

= (a

whence, eliminating rf/f/c/T,

)»}'+«»Arl - (24).

VOL. OCVII. — A. 2 Q

298

DR. L. N. G. FILON ON THE

To simplify the calculation, we may suppose that in the initial state of things
Q = 0, so that K = 0, p. = /AQ. As a matter of fact Q cannot be 0, but the assumption
that K = 0, p. = /AO is good enough to give the characteristics of the phenomenon
sufficiently well for our purpose.

We have then

In all the above d/dT! denotes rate of divergence with regard to T of a quantity
for the two oppositely polarized rays. Thus

= C,

= value of C if there were no absorption band.

Accordingly,

i -(xy/x)2} ^ {[i -

+ a>/x>} -i

gives the deviation produced in the stress-optical coefficient by the absorption band.
Calling this SO, we have

^•obs. '• \x\. = C + SC : C.

<w

ill!

\ \ \ \

I \

30

-

-

P 20

_

Ul

*X \fo

©

<t

** \

0

& 10

/ © \o

© __-;

1

/Q

~- " " ®

>

|o

f^s ^y ^

X" ^ .

y

G>"^ ©

©

^

©

X ^\

k-IO

v

/

M

*•*

®v

© ©/

CO~?0

—

\ '

-

|-30

-

V

-

1-40

\ \ \ \

, \

4500

5000

5500

T£NTH-METf?ES.

Fig. 10. Diagram showing curve of residuals from straight line for glass 3453.

Thus the deviation Xobs.— X^. which is given by fig. 10 is X8C/C.

X, /^o, C are all comparatively slowly varying : the factor which causes the
oscillation is

x'r. - ..... (25).

This factor starts with the value 0 when X = 0, decreases to a negative minimum
— X//a(2Xp+a) when X* = X//(X, + a), and then increases to a positive maximum
Xya(2Xp-a) when X2 = xy(Xp-a). It then decreases down to 1 when X = oo .

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 299

a being here small with regard to A,, the wave-lengths of minimum and maximum
arc approximately X,,— J.a, A., + \i. Neglect in;^ tin- variations of I In- otlirr tin -tors. this
result gives us an easy means of obtaining a. from fig. 10. a is the horizontal distance
between the maximum and the following minimum, a. is therefore between 300 and
400 tenth-metres — say 350 ; A, from the same diagram is about 4900.

This phenomenon also gives us experimental evidence in favour of a non-alteration
of the period. If we refer to the physical meaning of Q, we find it to be

Q = NeV/mTT,
where

T = period of the light corresponding to the absorption band,
N = number of electrons in unit volume which vibrate in this particular mode,

e = charge on such an electron,
m = its mass.

It follows that when we suppose T, that is A,, to vary (K being initially zero, as
before), what we have called 8C is given by

{Q [1 -

. . . (26).

Now, in the neighbourhood of A = A,, a being small, the rapidly varying terms
which determine the shape of the curve are those involving Aa— A,,8. If we call this
quantity x, and put A = \f in the other terms, we obtain some conception of the
shape of the curve. Taking

• ....... (27),

y has one minimum value — (a^A,,8)'1 when x = 0. It has two maximum values
(asAp*)~'/8 when x = ± v/3«Ay or A = \j,±\/3a.l2. The general shape of the curve is
shown in fig. 11. It is at once obvious that it does not show the alternate large
maximum and minimum required to fit the curve of fig. 10. So far, then, the
experiments bear out the hypothesis that the free periods are not altered.

Before we can proceed further we have to settle finally the convention about the
sign of C. It has been usual to call C positive for ordinary glass, such as that
investigated by BREWSTKR and KERR, and C negative for heavy flint like S 57.
This convention has been adopted by the author in previous papers.

We will now rigidly define the stress-optical coefficient as

C =|(MT,-MTl)/(TJ-T1) - ........ (28),

2 Q 2

300

DR. L. N. G. FILON ON THE

where /IT denotes the index of refraction of the ray vibrating along the direction in
which the principal stress (tension being considered positive, as usual) is T.

Since the direction of vihration — if we take the electric force to be the light- vector

—is perpendicular to the direction of polarization, and the index of refraction is

inversely proportional to the velocity, this means that a positive C implies a higher

i

-i-o

-5

-1-0

-\-5

-1-0

1-0

-5 0 -5

VALUES OF(\-kp)/a.

Fig. 11. Diagram of the curve y = [(X" - V)2 - <*2V] [(*2 - V)2 + "2V]~2-

velocity in the ray polarized in the direction of greatest stress. Now this ray
corresponds to the ordinary ray if the glass be compared with a uniaxial crystal
whose optic axis is parallel to the direction of greatest stress.

The glass, therefore, produces the same effect as a positive uniaxial crystal so
placed.

This is, in fact, what does occur. This definition, therefore, agrees with the
earlier one.

Now, since (for this glass) X, C, p.0 are all positive, it follows that if we are to have
a maximum followed by a minimum, as in fig. 10, dQ/dH < 0.

Now, in Q = NeV/wiTr, since r does not change, the only quantity which can
change with the polarization is N, the number of electrons per unit volume.

dQ/cTT is then eV(N3-N1)/7rw(T!1-T1), where N2, Nt are the number of electrons
of period T which respond to vibrations in the directions of T2 and TI.

Our result will, therefore, signify that a tensional stress decreases the number of
electrons which respond to vibrations in the direction of the stress, relatively to the
number responding to vibrations in a perpendicular direction. In other words, tension
appears to tend to set the electrons vibrating in a plane at right angles to the line of
stress, pressure having the reverse effect.

It seems probable that if this effect were due to an absorption band, a glass

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 301

showing in its unstressed condition strong absorption bands in the visible spectrum
would exhibit this effect in a most marked manner. A small slab of didymium glasp
was obtained and examined under direct pressure with the apparatus described in
the previous paper in ' Camb. Phil. Soc. Proc.,' already referred to several times. The
ol)servation8 were not very precise, and it is hoped to repeat them with the flexure
apparatus, when larger slal» can be obtained.* Such as they were, however, they
gave a negative result. The l>and due to stress passed through the double absorption
band in the orange without showing any marked irregularity.

It seems, therefore, that the didymium electrons which produce the alworption
band are not affected by the stress in the way described. It appears possible that
the didymium particles really float about, as it were, in the glass, in a free state, like
particles in suspension in a fluid, and that they cannot be influenced to any great
extent by stress applied to the glass. Further research in this direction is in
progress.

§22. Determination of Absolute Values of C.

Although the experiments were primarily undertaken to show the dispersion
effects, it is desirable to know also the absolute values of the stress-optical coefficients.
These are not given by the experiments as described in § 2, because the differences of
level «i— Zj and z^—h cannot be measured with sufficient accuracy.

To determine absolute values a second slit is used, of which the height is h + b.h.
The two slits were cut in the same diaphragm, so that A/4 is easily measured once
for all.

Referring to formula (7), the relative retardation due to the second slit,

where C', = stress-optical coefficient of beam F for the wave-length for which the
retardation is R'.
Also

R = (3M,C/2V)[>i-z,+<r(zi-A)] ....... (30).

Hence

RVC^-R/C^ -3M/TA///2V ....... (31).

•

If therefore we know the ratio CX, : C3 we can find the al«olute values of either.
If R', R correspond to kinds of nlb order,

R = nX, R' = nX'.
Therefore

nfX'/C'.-X/C,) = -3M,«rAA/26,a.

[*JVofe added April 3, 1907. — Since writing the above the experiment has Iwen repeated under the more
accurate conditions, and the negative result has been confirmed. If the effect exists in the didymium
glass it is certainly small.]

302 DR. L. N. G. FILON ON THE

Assuming the law connecting C and X to be given by (14), we have

C- r1 <*>//! \ w/i^ r" — P w//i \ <2>/x'\
j — \JQ f{l — Ay /A^, \j j — V>o /\A — Au ^A ^,

where G^, V2> refer to the slab F.
Hence

Therefore

and from equation (5),

/I

H,(X'-X)/C0(a>= -3Mao-A/t/26/.
Co(2)=-2n(X'-X)&//3MJo-A& (32),

>A (33),

Co(1) = 2n (X'-X) &!» (I + 1
\<r

which give the absolute values of C0 for both slabs.

A great many errors enter into the determination of these absolute values. It is
very difficult to measure the spans with sufficient exactness, and the differences X'— X
are not large enough to allow of very accurate determination.

§ 23. Effect of Chemical Composition on Stress-optical Properties.

The mean values for Crt obtained in this way for each beam are shown below in
Table VI. A, B denote the two individual slabs of each pair, and C0 is expressed in
a unit equal to 10~7 (cm.)2 per kilogramme weight.

TABLE VI. — Dependence of C0, X,, on Chemical Composition.

Glass.

C0 for A.

C0 for B.

Mean Co-

*»

BaOs.

K20.

B203-iK20.

3413

2-99

3-11

3-05

553

33-0

3-8

31-1

1809

2-95

2-94

2-94

561

34-3

7-4

30-6

935

2-82

2-94

2-88

451

27-7

3-1

26-1

3296

2-71

2-83

2-77

537

15-4

16-7

7-0

3749

2-15

2-19

2-17 "

564

5-9

23-9

- 6-1

3453

2-13

2-10

2-11

524

5-7

20-8

- 4-7

2783

1-93

2-21

1-93*

500t

1-4

12-5

- 4-9

* 1-93 has been taken, and not the mean of the two values, because here Cu certainly differs for slabs
A and B, and A was the slab analysed.

t Estimated from the values of Ar on p. 291.

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 303

If we refer to the diagram published in a paper previously referred to (' Camb.
Phil. Soc. Proc.,' vol. XII., p. 335) showing dependence of C upon percentage of PbO,
we see that until this percentage reaches about 40, PbO has little influence on the
stress-optical effect.

We are therefore to look at the two remaining principal constituents, Bj03 and
K8O, for the cause of variations in GO-

Looking at Table VI., in which the glasses are arranged according to descending
order of magnitude of C0, we see that the four glasses with the high Cu all contain a
percentage of B,03 which is considerable. On the other hand, the three glasses with
a low C0 all contain a comparatively high percentage of KaO. We conclude that
either BjO3 raises C0, or KaO lowers it, or both. Also, looking at 2783, which is the
lowest of the seven, we notice that it contains the least percentage of BjO3, but not
the highest percentage of K,O. This suggests that BjO3 is more efficient in raising
GO than K,O in lowering it. That the effect of B,O3 must be predominant is other-
wise evident from the fact that the order of the percentages of B,03 is the same as
the order of magnitude of C0, with one exception (1809).

The last column of Table VI. shows the values of (percentage of BjOs)— £ (percentage
of KaO). This places 1809 in the sequence, but throws out 3749. Also the marked
difference between the four first and the three last is shown very clearly.

The glasses do not form a sufficiently regular series to enable us to go further and
to determine exactly the law of dependence of Gu upon the percentages of BjO3 and
of KSO in the glass. But the increase of B.,O3 certainly increases C«, and the
increase of K2O probably decreases it; and Bj03 seems to be at least twice as
efficient as KaO.

With regard to the mean values of AO, they all appear to be of the same order of
magnitude. At all events no definite dependence of AO upon the composition can be
traced. For percentages of B,O3 not exceeding 35 and of KaO not exceeding 25 it
would seem that Ay is independent of the composition of the glass or that the
dispersion is, in every case, proportional to the double refraction.

§ 24. Failure of HOOKE'S Law for Glass 2783.

Before concluding, the peculiar phenomena shown by glass 2783 require explanation.

This glass showed a progressive increase with increase of load in rAA/AWr for
both tension and pressure.

Also when the load was increased the band, which was straight and vertical for
moderate loads, became curved, being convex towards the red as shown in fig. 12.
The load was eventually increased to about 59 kilogrammes when one of the glasses
broke. The baud was observed when the glass was on the point of rupture and it
exhibited a decided V shape as drawn.

304

DR. L. N. G. FILON ON THE

This showed that the relative retardation was greater for light passing through at
mid-level than for light passing through the edges of the beams.

Such an effect can be explained only in two ways. Firstly, by supposing that
the law of stress across the section does indeed remain linear, but that the relative

I/I OLE T

H/CH

Jusr

LOAD
Fig. 12. Appearances of band for glasses 2783.

retardation is not proportional to the stress, increasing less rapidly for large stresses
than for small, since here (the overlapping being assumed for simplicity to be J height
of beams) |T + |-T produces more effect than T.

Such an effect, however, would imply an increasing falling off of the values of A
from the values which would be obtained if the linear law held. The result would be
a progressive decrease, instead of a progressive increase, in rA\/AWr.

Secondly, we may suppose that HOOKE'S law fails or stress is not proportional to
strain. Consider a beam bent under constant bending moment. The axis will take
the form of a circular arc. If we imagine the circle completed, then the symmetry
shows that the cross-sections must remain plane and all pass through the centre of
the circle.

It follows easily that, whatever be the law of stress, if we can neglect end effects, the
extension follows a strictly linear law.

C IT

Thus if AB (fig. 13) represent the vertical axis of such a section, and the stress at
any point R in AB be set off as RP perpendicular to AB, the locus BPC of P is a

DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 305

genuine stress-strain diagram. It will therefore, as is well known, take the shape
shown in fig. 13, the stress falling off rapidly as the strain approaches and passes
what is known as the yield-point.

The corresponding curve for the overlapping part of the other beam is shown by
AQD, and the stress effective in producing the optical effect is the sum of the two
ordinates RP, RQ. It is obvious from the figure that PQ is a maximum in the
middle. Thus the peculiar shape of the band is likewise accounted for on this
hypothesis.

Now let the straight line BI give the stress-diagram which would be obtained for
the same bending moment if HOOKE'S law held. This straight line and the curve
must then be so related that the first moment about BX of the areas APCB and AIB
are equal.

Let L be the extremity of the ordinate through the mid-point of AB. Draw BL
cutting AC at J and the tangent at L cutting AC at T, BX at U. The stress-strain
curve is always convex inwards, therefore CPLB always lies on one side of TU as
shown.

Now the triangles BLU, TLJ are clearly equal. Hence area CPLJ > area BSL.
And the mean distance from BX of the area CPLJ > mean distance from BX of the
area BSL. Therefore first moment of CPLJ > first moment of BSL, or first moment
of ABJ > first moment of ABLC. Therefore, if ABI and ABLC have the same first
moment, ABI < ABJ, or BI must lie to the left of BJ.

If BI cut ML in K, then ML > MK. That is, the actual measured stress is
greater than the computed stress. Therefore the observed values of A. exceed those
that would be obtained if HOOKE'S law held by a difference rapidly increasing with
the stress.

The result is a progressive increase in the value of 7-AX/AW, such as is actually
observed. The discrepancies which have appeared are therefore explained.

Incidentally this confirms the conclusion (which indeed seems highly probable on
theoretical grounds) that the stress-optical effect is dependent upon the stress — that
is, the molecular strain — and not upon the molar strain. The latter, which is the
sum of both plastic and elastic effects, is the quantity measured in most extension
experiments, and is usually denoted by " strain " simply.

§ 25. Conclusion.

This completes the account of the results reached so far. The next step would be
to obtain glasses of suitable chemical composition to show the effects discovered in a
much stronger degree and thus allow of more precise determinations. Research in
this direction is being undertaken, and it is hoped that the results will form the
subject of a future communication.

VOL. CCVII. — A. 2 R

306 ON THE DISPERSION IN ARTIFICIAL DOUBLE REFRACTION.

In conclusion the author wishes to express his most grateful thanks for the
assistance which has been rendered him by the Royal Society, without which this
work could not have been undertaken. He desires also to place on record his
indebtedness to Professor F. T. TROUTON, F.R.S., who has most generously placed the
resources of the Physical Laboratory of University College, London, at his disposal ;
und also to Mr. A. W. PORTER, B.Sc., whose interest in the work and kindly criticisms
and suggestions have been invaluable.

[ 307 ]

VIII. The Dititriliiitiun <>f nine-Violet Light in the Solar Corona on August 30,
190;"), as derived from /'/in/ot/ra/i/ix taken (it K<il(ta-es-Senam, Tunisia.

By L. BECKKK, f'/i.D., Rcyiux I'rofi'ttsor of Astronomy in the University of Glasgow.

Communicated by the JOINT PERMANENT ECLIPSE COMMITTEE.
Received November 27, 1906, — Read June 6, — Revised August, 1907.

[PLATE 1.]
CONTENTS.

Section Page

1. The apparatus ........................... 307

•>. The photographs ........................ 311

3. The measurements ..................... .312

4. Kfdui'tions ............................ 313

5. Correlative distances ........................ 315

6. Photographs VI. and VII ............ . . ........... 319

7. Intensity formula .......................... 321

8. Formula and position angle ...................... 328

9. Numlier of particles and intensity of light ......... ..".... . 329

10. Plea for repetition of observations .................... 331

Appendix I. — Diffraction due to screens ................ . 332

„ II. — Comparison of corona and moon ............... 335

Tables I.-IV ............................ 337

Talile I., $ 4 (>•), and -^ 7 and S con tain the results derived from the measurements made on
.the photographs, viz., equal-intensity curves of the corona and a formula (D) expressing
the intensity of the corona as a function of the mean distance from the solar limb of an
equal-intensity curve.

§ 1. The Apparatus.

The object of my expedition to Kalaa-es-Senam, Tunisia, was to obtain a series of
photographs from which might l>e determined the distribution of light in the corona.

In designing my apparatus, I was led by two considerations: (1) the photographs
had to be taken automatically, as I had to work without assistance, (2) standardising
of the photographs was to be avoided. All the photographs were therefore taken on
the two halves of a whole plate placed end to end and developed in the same tray
during the same time. The automatic apparatus gives 10 exposures, and it is
governed electrical Iv 1>\ a pendulum dock. I employed two cameras, one with a

VOL. OCVII. — A 420. 2 R 2 19.12.07

DOS

PROFESSOR L. BECKER ON THE DISTRIBUTION OF

Cooke triple achromatic lens of 3£ inches aperture and 58-5 inches focal length, which
belongs to the Glasgow spectrograph, the other with a Ross portrait lens of 2 inches
aperture and 12 inches focal length. The pictures obtained with the larger camera
are so much superior to the small size ones of the portrait lens that I have not made
use of the latter in this paper. The cameras were fed by a coelostat of 8 inches
aperture, which had been kindly lent to me by the Royal Dublin Society. In front

Fig. 1.

of the two object-glasses, and about an inch from them, a rotating shutter was
mounted which served both cameras. The rotating shutter has four oblong apertures,
90 degrees apart (its back view is shown at D2, fig. 1) ; it is rotated by clockwork
driven by a spring, and its motion is governed by the armature of an electro- magnet (/).
When the armature is attracted, the shutter rotates through about 45 degrees until
it presses against one of the four stops d and brings an opening opposite the object-

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 309

glasses, and when the armature is released the shutter turns again 45 degrees, as far
as one of the stops c, and shuts off the light. The contacts are made by a pendulum
clock, and they are so devised that make or break can occur only when the pendulum
is at or near its position of rest.

I arranged for five exposures each of 1 second duration, and five exposures lasting
respectively 3, 9, 20, 46 and 89 seconds. Their actual durations are 0'84, 0'80, 078,
0'80, 0'85, 2'82, 9'02, 20'84, 45'91 and 89'04 seconds, as determined automatically on
the chronograph at the Observatory after my return from the eclipse expedition. I
have deducted 0'02 second from the figures recorded on the chronograph to allow
for the peculiar motion of the shutter. At the first four exposures of 1 second,
different screens, each with 13 holes, are in front of the object-glass. The diameters
of the openings are respectively 0'210, 0'296, 0'410, 0'595 inch. At the first exposure
the screen leaves 1/2T4 of the object-glass free, at the second 1/10-8, at the third
1/5-6, and at the fourth 1/27. These screens are geared to the clockwork which
rotates the shutter and fall out of gear after the fourth exposure. The illustration
shows them out of gear.

The plate-holder (C) of the Cooke camera is 17x3 inches; it slides lengthways
inside a metal box 32 x 4 inches. It is moved by rack and pinion, the rack being
attached to the plate-holder, and the bearings of the axle of the pinion to the cover
(C,) of the box. Spring-driven clockwork (B) communicates its motion by means of a
shaft (a) to the pinion. The clockwork is governed (at b) by the armature of an
electro-magnet (the armature and the revolving stop with its axle appear white in
fig. 1). When the armature is attracted, the plate-holder moves 1 inch onwards, and
when it is released it moves another inch. The necessary contacts are made by the
pendulum clock. I arranged the contacts in such a way that for the first four
exposures the plate moves one step onwards, for all the others two steps, and when
the plate has l>een pushed along, 2 seconds are allowed for the camera to settle
before the next exposure is made. Of the 206 seconds for which I made provision,
173 seconds are occupied by the exposures, 15 seconds are taken up by changing
of plates, and 18 seconds are lost.

The pendulum clock is shown at A. It is provided with four circular steel-sheet

discs, into which notches are cut. The axle which carries the discs has a period of

240 seconds, i.e., about half a minute more than totality lasted. Two of the discs

(A, B) are represented in the diagram, fig. 2, which

J ~"~"xi eiU a^80 shows the contact levers. The diagram gives

•"'A ^: k-|l' — __ the position immediately before making of contact.

At the next second a will fall on b, making
contact at c, and after another second b will fall
away from a, thus breaking the contact. The

duration of contacts depends slightly on the position of the notches, as will lie seen
from the figures given above for a second's contact, which show a range of 0'07 second.

,310

PROFESSOR L. BECKER ON THE DISTRIBUTION OF

It can be shown from the observed durations that the clock was off-beat at the
Observatory, and probably it was so, too, at the eclipse, and possibly in the opposite
way. This would not affect the relative duration of the first five contacts, as they all
lie between an uneven and even second, but their errors would appear relative to the
long exposures.

The sequence of events governed by the clock takes place at the moments of time
shown in the following table, where the numerals denote the seconds elapsed from
second 0, when the pendulum is started :—

Contacts

Contacts

for exposure.

for change of plate.

.Make.

Break.

Make.

Break.

1

2

1. Exposure of 0-84 second, first screen, r = 77.

3

Plate moves 1 inch.

5

6

2. Exposure of 0 • 80 second, second screen, r = 55.

—

7

Plate moves 1 inch.

9

10

3. Exposure of 0'78 second, third screen, r — 39.

11

Plate moves 1 inch.

13

14

4. Exposure of 0 • 80 second, fourth screen, r = 27.

—

15

Plate moves 1 inch.

17

18

5. Exposure of 0 • 85 second, full aperture, r = 1 6'7.

19

20

Plate moves 2 inches.

22

31

6. Exposure of 9 • 02 seconds, full aperture.

32

33

Plate moves 2 inches.

35

38

7. Exposure of 2-82 seconds, full aperture.

39

40

Plate moves 2 inches.

42

131

8. Exposure of 89 • 04 seconds, full aperture.

132

133

Plate moves 2 inches.

135

156

9. Exposure of 20 • 84 seconds, full aperture.

157

159

Plate moves 2 inches.

1G1

207

10. Exposure of 45 -91 seconds, full aperture.

r designates the ratio of the focal-length and the diameter of a lens, which has the
same area as the lens reduced by the screen.

There are several points in the design of the apparatus which have proved
unsatisfactory. The shutter must have a smaller moment of inertia, and its motion
should be recorded on a chronograph ; the plate-holder ought to run on wheels
instead of sliding on a rod. The mutual distances of the pictures ought to be, say,
four solar diameters, and, especially, the side of the square opening in front of the
plate must be twice as great as the distance between the pictures [see § 5 (/), (g), (/<•)].
One of the pictures (not the last) must be 8 diameters from its neighbours [see § 4 (c)].
The screen for cutting down the aperture of the lens ought not to contain a series of
small openings, but have a central opening and an annular opening, whose diameter is
about two-thirds of that of the lens (see Appendix I, p. 332).

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 311

§ 2. The Photoijraphs (Plate I ).

The pendulum was started about a second after Mr. H. MAYOR, who watched the
contact, gave the signal that totality had l>egun. Before the last exposure was
finished sunlight apju-.-in-d, hut I shut it oft' by stepping in front of the object-glass,
and it was about 3 seconds before the shutter automatically closed. That is to say,
totality lasted about 1+207 — 3 = 205 seconds, as compared with the calculated time
of 210 seconds. Accordingly, the tenth exposure lasted about 43 seconds.

The plates (two halves of a whole plate) were developed together in the same tray
by a strong developer (Imperial standard) for 7 minutes. I developed at the open
window at star-light, keeping the plates covered most of the time. The photographs
show a great deal of contrast, and this has proved an advantage in measuring them.
The background of the long-exposed negatives is dense, due to the brightness of the
sky. This diffused light, whose intensity I had underrated in the design of the
apparatus, produced an impression even for the shortest exposures, darkening a square
on the plate equal in size to the opening in the plate-holder. I find, from measure-
ments, that the intensity of the diffused light equals that of the corona at a distance
of I'l solar diameter from the sun's limb, i.e. 0'6 in unit of the intensity of the corona
at a point 1 solar diameter distant from the limb as found from the formula § 7.

In the preliminary report I have said that the plate-holder failed to move in the
designed manner, due to some parts of the apparatus having been damaged in
transit. Owing to this accident, there is a multiplication of images in the sixth
and seventh pictures, and not only the first five photographs are an inch (2 solar
diameters) apart, but also the next two, which were meant to be at twice that
distance. In consequence, the successive exposures to the diffused light overlap on
the plate, with the effect that on one half of each of Photographs I. to VII. there is
the same duration of exposure to the diffused light as on the adjoining half of the
neighbouring photograph. This has enabled me to separate the intensity of the
corona from that of the sky. There is no overlapping on Photographs VIII. and IX.,
and though they did not, on that account, furnish data for the intensity formula, they
supplied H series of equal-intensity curves of the corona, which are required for the
reduction of the other pictures. Fig. 3 explains the conditions.

I - H ta

(

) (

) (

) (

) (

) & (

>

o

0

i n m B v

IX

Fig. 3.

The photographs are numbered I., II., &c., and the exposures (see § 1), 1, 2, <kc.
The lines at the top give the extent of the area illuminated at each exposure by the

312 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

diffused light. The left aud right halves of a photograph will be designated by a
and b. Photographs I. to VII. occupy the first half-plate, and VIII. and IX. half of
the second half-plate.

Photographs VI. and VII. are, from an ideal point of view, marred by defects —
No. VI. by some instantaneous pictures of the protuberances which appear on the
lunar disc, and No. VII. by two short exposure pictures which are eccentrically
superposed on it. These defects are of no consequence (see § 6). Owing to the
failure of the automatic apparatus, the exposures of Nos. VI. and VII. are uncertain
to about a second, but the sum of the two exposures, which equals the sum of the
sixth, seventh and eighth exposures, is accurately known.

To get the negatives, from which the Plate was prepared, I first made enlarged
positives, copying those pictures together whose background have the same density
on the original. I have attempted to make the coronas of Via. and VI6. extend
equally far, and in the attempt the coronas of pictures V«. and VIII. have come out
too small. I should say that on the original negative the corona of VIII. covers the
whole breadth of the plate, which towards the top is three-quarters of a solar
diameter broader than shown in the reproduction. The enlargement is 1*7.

§ 3. The Measurements.

The observations made on the photographs, and utilised in this paper, consist in the
selection of points on the several corona pictures at which the photographic film shows
the same degree of blackness, and in the measurement of their distance from the lunar
disc. The measurements were actually made on positives, and not one but twenty-
four points of an equal-blackness curve, 15 degrees apart, were measured. The
positives are contact prints on slow plates obtained at a distance of 10 feet from a gas
jet. The twelve sets which I prepared belong to different exposures, and were
developed for contrast. I copied the negatives I. to Va., Vt. and Via., VI6. and VII.
separately on account of the differences in density of the background. Sixteen sets of
measurements of equal-density curves were made on these twelve sets of positives.
The measurements were easy to make, and proved to be consistent. The positives
show a perfectly transparent ring round the lunar disc, the diameter of which depends
on the exposure and development. Seen against black paper, this ring furnishes a
well-defined outline to set upon. Some of the curves at great distance from the sun,
where the intensity changes little with the distance, were measured on enlargements
(10 diameters) on bromide paper, in which the contrasts are much increased.

I further made twenty copies of each of Photographs VIII. and IX. at all kinds of
exposures. These negatives are very dense, and, apparently, evenly dense up to
about half a diameter from the sun's limb, and show no detail to the eye when
inspected against a strong light. On the other hand, the positives contain detail as
near as 0'12 diameter, and as distinctly as if they were replicas of the first six

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 313

negatives. Tli«- miM^im-nu-iits tiiinMu-<l 7'.» DOTW »f e,|ii.il il.-iisily. \\liirli !„•!,, n^ t<>
mean distances between 0'12 and 1 solar diameter. As each of the two photographs
has a background of equal density all round, these curves will be employed in reducing
measured distances on other photographs as described in § 4 (c) and § 4 (d).

The apparatus which I employed in measuring the photographs consists of a low-
power microscope mounted on a slide whose position can be read by vernier to O'OOl
of an inch. The slide is mounted on a circular plate which turns in a ring, so that
measurements can be made at any position -angle.

The measurements were made at position -angles 0 degree, 15 degrees, <fec. The
position-angles refer to the north pole of the sun. I obtained the zero of the position-
angles from the calculated position-angles referred to the north pole of the sun and
the positions of the second and third contacts on the first and last photographs. The
position-angles of the contacts referred to the celestial pole were found from the data
given in the 'Nautical Almanac' (124 degrees and 287 degrees), and the position-
angle of the north pole of the sun is 207 degrees.

On the photographs published with this paper the line joining the centres of the
lunar discs has a position-angle of 239 degrees, 59 degrees being to the left and
149 degrees at the top.

1 observed the following rule in measuring : — After clamping the microscope at a
certain position-angle, 1 set the wire successively on the moon's limb, then on a point
of the corona where the blackness had a certain density and, without turning the
microscope, made similar measurements 180 degrees from the first position. Keeping
the degree of blackness in my mind, I repeated the operation on the other photo-
graphs, and then for the other position-angles.

•
§ 4. Reductions.

The object of the reductions is to find (l) the mean distance from the solar limb of
each equal-density curve, and (2) the position of each equal-intensity curve with
reference to its mean circular-intensity curve. The steps are as follow :—

(a) There is a slight difference amounting to a few thousandths of an inch between
the diameters of the moon as obtained from negatives and from positives. On the
negatives the lunar diameter is 0'565 of an inch, and I reduced all the measured
distances to this diameter by correcting them by half the difference between this
figure and the diameter appertaining to each measured distance.

(b) Ifi'i/ii'-fion of the Distances front Lunar Limb to Solar Limb. — M (fig. 4) is the
centre of the moon, A, B and S are respectively the centres of the sun at second and
third contacts and / seconds after the second contact. The duration of totality is
about 205 seconds. The diameter of the moon is D = 0'565 of an inch, and that of
the sun is d = 0'540 of an inch. The angle between the second and third contacts,
1567 degrees, is given by the first and last photographs; it equals the angle

VOL CCVII. — A. •_' s

314

PROFESSOR L. BECKER ON THE DISTRIBUTION OF

subtended by BA at M. The position angle, counted from tbe north pole of the sun,
of the second contact is 104 degrees. CMS is designated by a.

A measured distance m, at position-angle P, is reduced to distance h from the sun's
limb by the following formula, small quantities being neglected,

tan a

./I- *
V 103

tan 7 8° -3, /t =

125 1-

cos 7 8° -3
cos a

cos (P

-104-78)1.

The maximum of h— m is 0'025 inch. The correction of the position-angles is
inappreciable for our purpose.

(c) Curves of Equal Intensity of the Corona. — I define the mean distance of an
equal-intensity curve of the corona as the mean of the distances of twenty-four points
of the curve, 15 degrees apart. Equal-blackness curves coincide with equal-intensity
curves on Photographs VIII. and IX., and also on Photograph I. The measured
distances were first corrected for corrections (a) and (6), and then each twenty-four

Fig. 4.

Fig. 5. '

distances belonging to an equal-intensity (or blackness) curve were combined to a
mean ; the differences, 8A, (mean minus reduced distance) define the equal-intensity
(or blackness) curve with reference to the circular mean curve. Finally curves were
interpolated from the observed 95 curves at regular intervals of the mean distance.
An extract of the results is contained in Table I. (p. 337), and graphs of some of the
curves are shown in fig. 5.

(d) Reduction of the Distances of Portions of an Equal-blackness Curve to the
Mean Distance of that Curve. — Though equal-blackness curves were measured on all
pictures at all position angles, only portions of these curves can be used together,
because the equal-blackness curves do not everywhere coincide with the equal-
intensity curves of the corona. In next section it will be shown (l) that in the case
of Photographs V., VI, and VII., owing to luminosity of the sky, the left and right

BLUE -VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 315

halves of an equal-blackness curve coincide each with an equal-intensity curve of the
corona, though not with the same curve of the corona, and that on the two halves of
Photographs 1. to IV. an equal-blackness curve differs inappreciably from an- equal-
intensity curve ; (2) that on account of the overlapping of the coronas belonging to
neighbouring pictures the intensity and blackness curves do not coincide at certain
position -angles. Therefore, if the mean distance of each equal-blackness curve be
derived separately for each half of Photographs V., VI. and VII., and all measure-
ments be excluded which belong to position-angles where there is appreciable
overlapping of coronas, the mean distance will also be the mean distance of an equal-
intensity curve. Each measurement belonging to a position-angle p, and reduced in
accordance with (a) and (6), plus the correction oh derived above (c), gives a mean
distance of the equal-blackness curve, and there are as many values of this mean
distance as there are measurements. Their average value h is the final value, and
its error can be determined from the differences from the mean. Table III. contains h,
its error and the number of measurements p which contribute to the mean value.
p = 24 indicates that all the points of the curve, position-angles 0 to 345, were used.
For p = 21, the points at position-angles 225, 240, 255 degrees are excluded and for
p = 19 those at 210 and 270 degrees are omitted in addition. For Photographs
V6. to VII. the points omitted lie symmetrically round position-angles 60 degrees
and 240 degrees. The quantities are given in unit of 10~3 solar diameter, those
derived in inches being multiplied by 1'852 (diameter of moon on photograph 0'565
of an inch, diameters of moon and sun 994'5 and 9507 seconds). I designate by
" corresponding distances " the distances from the sun's limb of two points on two
different pictures of the corona at which there is equal blackness. In Table III. the
mean corresponding distances stand on the same line. I shall show in the next section
that at these tabulated distances the ratio of the intensities of the corona is a constant
for each two photographs.

§ 5. Correlative Distances on Corona. [Definition see under (</).]

(a) I employ the following notation. 1 or S is an intensity of light acting on a
photographic plate, and they are the quantities of light falling on unit area of the
plate, which is the area cut out in the focal plane of the camera by unit of spherical
angle at the centre of the object-glass, i or s is an intensity of a luminous object,
i.e., a quantity of light falling from unit area of object (area cut out by unit of
spherical angle) on unit area of the object-glass, which unit area equals that for the
plate, a designates the exposed area of the object-glass and t the time during which
the plate is illuminated. Then I = as and S = as.

For the pattern of screens by which I reduced the aperture of the lens the loss of
light due to the object-glass will be about proportional to the aperture, and it need
not be taken into account, but the effects of diffraction require special investigation
(see Appendix I.).

2 s 2

316 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

(b) Intensity and Time for Equal Blackness. — Experiments have proved that for
the same photographic plate and development the intensities of light and the
durations of exposure necessary to produce the same blackness on the film bear a
certain relation to each other. According to MICHALKE this relation is independent
of the degree of blackness. I have represented MICHALKE'S observations by the
formula 1 1" = constant, where a= T08, and further redetermined a for the plates
employed by me (Imperial special rapid). I used as a source of light a disc of opal
glass (2 inches in diameter), illuminated from behind by an electric lamp, and I
exposed directly to the light of the disc successively different portions of the same
plate at distances varying from 1 to 15 metres. A Thornton-Pickard shutter
recorded the duration of exposure automatically on a chronograph. My value of a is
1-05 ±0-01.

(c) The ratio of two intensities, im, in, which illuminate, through apertures am, a» ot
a lens during times <M, tn, a photographic plate placed in the focus of the same camera
is a constant if they produce equal blackness on the film. By (a) and (b)

i~n=am \tj

FmH can be calculated for the eclipse photographs. The individual exposures of
Photographs VI. and VII. are uncertain to about a second (see § 6), but their sum is
accurately known (100'88). I take here t6 = ll'OO, and hence t-t = 89'88. The other
data are given in § 1. The numerical value of a is of no importance for the first five
photographs.

m, n 1,2 1, 3 1, 4 1, 5 5, 6 6, 7.

log F™ 0-278 0-551 0-884 T335 1'059 0'869.

(d) Correlative Distances on the Corona. — If the pictures of the corona had not
been overlapping, and the sky been dark, an equal-blackness curve would have
coincided with an equal-intensity curve of the corona, and the ratio of the intensities
of the corona belonging to two such curves on Photographs m and n would equal a
constant FmB [see (c)]. I shall call "correlative distances on corona" the distances of
points of the corona at which the ratio of the intensities equals Fmn.

(e) Simultaneous and Successive Exposures. — I make the following two assump-
tions : — (1) the degree of blackness on the film is independent of the order in which
two or more exposures are made ; (2) if two intensities give the same blackness for
certain exposures, they do so, too, when these exposures are made on an otherwise
exposed film. I have checked (1), but not (2), by experiment.

Let two intensities I and S illuminate the film together during the same time t.
By (b)

(I + S)"« = S'(t + t') = Sa* + S"f' for equal blackness,

BLUE- VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 317

where t' can be determined from the equation. The formula expresses that (I + S)
acting on the film during t gives the same blackness as S acting during t and if,
t' being, of course, after (or before) t. Hence the positive sign stands for " the one
exposure after the other." Each of the terms may be replaced by a term of the form
amb, which equals it in value [see (2)], and a expresses the intensity, 6 the time. The
terms may be written in any order [see (1)].

(/) Elimination of the Diffuted Light oj the Sky. — Let t and I belong to the
corona, * and S to the sky. On the second half of the mth photograph I«-fSm
illuminates the film during tm, and thereafter Sm+l during tm^ ; on the first half of the
(m-f l)th photograph SM illuminates the film during /„, and thereafter !„.
during £„+,. Let both produce equal blackness. By (e) •

+S-,*,^, = constant = S\A.+(I.+i+S.+,)'<.+,

The last two terms disappear, and therein lies the advantage introduced for Photo-
graphs V. to VIII. by the failure of the mechanism during the eclipse ;

1'V., = I\+,«.+l, where J' - I[(j+1)"- (?)"]

Substitute i and * [see (a)] and introduce F by (c) ; therefore

A = F. where .'„ = im [7?=+ lY- feYT".

«'.+i LW / WJ

Equal blackness was observed at the distances Am and Am+i, hence hm and AM+, are
corresponding distances ; they are, however, not correlative distances on the corona,
because the ratio of im and im+i, the intensities of the corona at /<„ and Aw+l, does not
equal F'w m+i- On the other hand, the distances hm + &hm, AW+1 + AA1I1+,, at which the
intensities of the corona equal i'M and i'm+i, are, by definition (</), correlative distances
on the corona. Hence we have

I calculate log i and the differential quotient by the formula derived in this paper,
which gives i as a function of h ; further, i' for « = O'fi, and thence A/i. The values
are: AA = 0 for h = 200, AA = -3 for h = 600, AA = -15 for h = 1000, and AA = -35
for h = 1400. The measured corresponding distances are correlative distances on the
corona with an error AA. These systematic errors are insignificant compared with
the accidental errors of measurement (see Table III.) up to A = 800, and even for the
most distant parts of the corona they do not reach these accidental errors. The
correlative distances determine the intensity formula (see § 7), and in the equations
the residuals appear under the form r = &hm— F1'4.-,, „ AAW_!. I observed on

318 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

pictures VI6. and VII. the corresponding distances A6 = 550, h7 = 1000, and /*„ = 900,
k, = 1400; hence v equals -10 for & = 1000, and -17 for h = 1400, while the
accidental errors v are about 45 and 75, that is to say, four times as great as the
systematic errors. Compared with the actual residuals v left by the equations,
the systematic errors are still smaller. The result, then, is this : Let there be equal
blackness on the two adjoining halves of two neighbouring pictures (Nos. m and m+ 1)
of the corona at distances hm and hm+l, then hM and hm+l are also correlative distances
on the corona at which the ratio of the intensities equals F,,,_ ,,1+1.

The result would have been different if the backgrounds of the two neighbouring
pictures had not been overlapping, i' = i+s would have been found instead oft, and
s could not have been separated from i.

(g) The intensity of the diffused light of the sky can be disregarded on the first
five photographs. To prove this, I start from the equations [see (/)]

(Im+Sm)aim-fSam±1<m:tl = constant for equal blackness.

The lower sign belongs to the first half of the mth photograph, and the upper sign
to the second half. For the first five photographs s is small compared with i, hence
by (a), (b), and (c)

«"« (im+ o-ra)%n = constant, where <rm = s (1 + FMi m±1)
*•£ = Fnm, where i'p = ip + a-p.

* n

<rm is the intensity which produces at aperture am and exposure tm the same
blackness as the two superposed exposures to the diffused light. The distances
belonging to the intensities i'm and i'n are, by definition, correlative distances on the
corona. We obtain, then, in the same manner as explained in (f),

fdh\
artm = <r, '

and

v =

<rm is calculated by the above formula and s = 0'6, except for the first photograph
and the first half of the second. The values range between 0'8 and 2 '3.

A minute before the beginning of totality the cap was removed from the object-
glass and during that time light must have been reflected into the camera by the
shutter, which was placed about an inch from the object-glass, and illuminated the
plate at the place where Photographs I. and lie*, were taken. The blackness of the
background lies between that of IV. and V., and I estimate tr, = 10 and cr2 = 5.

The calculated values of v do not amount to a third of the accidental errors v of
measurement (Table III.). It is, therefore, permissible to regard the corresponding

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 319

distances measured on pictures I. to Va. as correlative distances on the corona, just as
if no diffused light had been illuminating the plate.

After the experience gained at this eclipse I should again place the picture* as
they appear in the diagram, i.e. make the opening of the plate-holder twice as long
on a side as the distance between the pictures. This arrangement entails no
disadvantage for the short exposures, and for the long exposures the intensity of the
diffused light can be eliminated. (See § I, last section.)

(h) Overlapping of Coronas of Neighbouring Pictures. — At a point A of the mth
picture the intensity of the corona is (im) at distance hm, and the intensity of the
light which illuminates A for a time tM is (!„,) = «„,(*„.)• The same point is illuminated
for a time tmil, also by light of intensity (I,,±i) belonging to a different part of the
corona in the (m±l)th photograph, where the corona has an intensity (im±i) and
(I»±i) = a»±i (imii)- Point A lies on an equal-blackness curve of the m'h picture, and
this curve coincides with an equal-intensity curve of the corona (intensity = im) at all
points where there is no overlapping. By (c)

[o.t'.J1 tm = [a. (»„)]• tm+[amtl (»„,)]" «.±i,
or very nearly

^ = (»'.) +(t.±i)F...±l.

The equal intensity curve (intensity = im) cuts the radial line belonging to A
(distance = h) at A', and AA' = AA is the distance of the two curves at A. Hence

I measured on a diagram the distances of a point A from the solar limbs of the
following and preceding pictures, calculated im±l by the formula i = /(/') and thence
AA. In deriving the mean distance of an equal-blackness curve [preceding section (d)]
I used only those measured distances for which the average value of AA (including
AA = 0) is less than a half of the calculated error of the average distance. The
number of values is given in Table III. under heading p [see preceding section (t/)].
The average values of AA increase with the accidental error, but they have always
the same sign, so that the systematic residuals t; become very small compared with
the accidental errors t'. It would of course have been better if all the images had
been further apart. (See § 1, last section.)

The outcome of the discussion given in this section is, that the mean corresponding
distances given in Table III. are also mean correlative distances on the corona.

§ 6. Photographs Nos. VI. and VII.

(a) Duration of Exposure. — Owing to the failure in the driving of the plate-
holder only two pictures (VI. and VII.) belong to the three exposures 9'02, 2'82,
89-04 seconds. The sum of the durations of exposure of these pictures is thus given

320 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

(100-88). It is of some importance to know the upper limit, if not the accurate
durations, of the exposure belonging to Photograph VI. On Photograph No. VII.
three images are eccentrically superposed (see fig. 6). The order of magnitude of the
time for which they were illuminated can be found from the degree of blackness of

the background which belongs to each image. In this
way I find that circle (b) is due to a short exposure
of the order of a second, the faint circle (c) (dotted in
fig. 6) which is faintly visible within the corona is the
lunar disc during a very long exposure, and semi-
circle (a) must belong to an exposure of about
10 seconds. Now (b) is exactly at the position on
the plate, as determined by its distance from other
pictures, at which the plate-holder was locked by the
electromagnet of the propelling mechanism. The
contacts made by the clock for unlocking and locking
the plate-holder are between the contacts for exposures,
and hence no part of the exposure which produced (c)

can have contributed to (b). As (c) belongs to the longest exposure, (b) must be due
to some portion of the 2'8-seconds exposure. The first part of this exposure must
have contributed to picture VI., because the plate moved (see the instantaneous
photographs of the protuberances on the lunar disc of picture VI. and in the coronas
of VI. and VII.) while an exposure was going on. The exposure given to No. VI. is
therefore 9'02 + 2'82— T = 1T84— T, and the combined exposure for abc of No. VII. is
100'88, less the exposure of No. VI. T may lie between O'l and 1 second (see above).
I should add that the lunar disc on Photograph VI. is exactly round and not blurred
in the least. Apart from the instantaneous pictures and trails of prominences on the
lunar disc, the picture VI. is perfect.

(b) Measurement of Curve of Equal Blackness on Photograph VII. — I shall now
investigate whether the three eccentrically superposed images on this photograph can
be utilised in this research. Let the curves of equal intensity of the corona be circles,
and let us consider the curves at some distance from the sun. Let C and A (fig. 6)
be the centres of two photographs of the sun (not moon, as it was supposed in No. Ga)
and MM' and NN' be circles of radius r along which the corona has an intensity i.
To centre C belongs the exposure ^ and to A the exposure ta. Along circle MM', i has
been exposed during tt and on this is superposed an exposure to i— (di/dr) Ar during
ta, i.e., i has been exposed during t^ + t, and — (di/dr) &r during t2. This additional
radiation has the same value with opposite signs at two opposite points of circle MM'.
In the same way along circle NN' intensity i has acted during ti + t3, and (di/dr) Ar
during ta. Hence the radiation i was exposed during t} + 12 along a curve which lies
between the two circles MM' and NN' and in such a way that the mean of the
distances (p and p') from C of two opposite points of this curve is equal to r. Along

HI. UK-VIOLET LIGHT IN Till sni.AR CORONA ON AUGUST SO, 1905. 321

this curve there is equal blackness on the photograph and the same blackness occurs
on Photograph VI. at a point at distance h«, which was illuminated by »'„ during t6.

Hence

/«/i = constant = («, + f,)1 '•/««v'.

The mean distance of the equal-blackness curve is the average value of p which
is r. But r is, by assumption, the distance of the point on the corona at which the
intensity is i. Hence the menu distance of the equtil-hliickness curve and ht are
correlative distances on the corona. The same result holds good for the three
eccentrically superposed pictures ftbc of the VIIth photograph, and the constant F,, 7 is
equal to (ti + fa + fx)l"/ta". Photograph VII. may therefore be measured and reduced
in the same way as the other photographs, provided that always two opposite points
of an equal-blackness curve be measured. Terms of the second order have been
neglected in this derivation ; they amount to only a fraction of the distance AC
(O'll diameter) and are small quantities compared with the errors of measurement.

§ 7. The Formula, which gives the Intensity of the Corona as a Function of the

Distance h.

I first tried whether the observed distances satisfied Professor TURNER'S formula
(intensity inversely proportional to the sixth power of the distances from the sun's
centre), but find inadmissible residuals. Another formula has therefore to be derived.
If the distances given in columns I. to Vre., Table III., be plotted as ordiuates, and
the corresponding distances standing in the first column as abfccissae, the points
belonging to the same column lie as nearly in a straight line as can be expected from
the accuracy of the observations, and all these five lines can be made to intersect in a
point —x, —x.

Hence

= yn(hf + x), n — 1 to 5, x a constant.

The intensity * being a function of the distances h, which are counted from the
sun's limb, I write i = cf(h+x). Hence »', = cf(hi + x) = </[y. (A.+x)] and
*» = </(/»„ +x),

'i = constant F.

as /», and hn are correlative distances on the corona. This relation is satisfied by
j(z) = z~*. Hence i ' = c(li + jc)~* and Ft , = y»~y- The formula is the same as
Professor TV UN KK'S, with this difference, however, that x need not be the radius of
the sun.

Approximate values of x and y are found in this way. I assume x = 0, 40, &c., to
320 (solar diameter = 1000) and calculate y« from hi, hm, and x. The residuals are

VOL. ccvii. — A. 2 T

322 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

v = ht + x-yn(hn+x). I take the sum of the residuals irrespective of sign for n = 2
to 5. For each value of n the sum of the residuals is a minimum for x between 80
and 120, and the sum for all values of n together is a minimum for x = 110. The
sum of the residuals is 50 per cent, greater for x = 20 and x = 300. x = 500 (radius
of the sun) leaves inadmissible residuals.

The observed values of FI „ and yn belonging to x = 110 give y = 3 '4 for n = 2,
3'3 for n = 3, 3'5 for n = 4, and 3'8 for n = 5.

In the same way, if the distances obtained from Photograph No. Via. be plotted
as ordinates and those on Photograph V6. as abscissae, a straight line represents the
observations as well as any smooth curve that can be drawn, and the same remark
refers to the distances obtained from Photographs VI£>. and VII. The sum of the
residuals is again small for values of x near 110, though the range of possible values
of a: is larger. The resulting value of y lies near 4.

I think this is sufficient proof that the function represents very nearly the
observations. Let us assume it to be exactly correct. The method explains that x
and the n constants y are determined independently of the times of exposure from
the condition h + x/^ + x = h\ + xlk'n+x — yn, and that there are n equations for the
unknown y. These n equations will be rigorously satisfied, provided the correct
values of F be introduced, and hence n—l values F,,n or n — l values of the time of
exposure can be determined from the equations along with y.

I prefer to determine x and y together by the Method of Least Squares. Let
#o+f, 2/o +ij be the true values of x and y, and vn be the accidental error of measure-
ment of hn, and AFMn the correction of an approximate value Fmn, which need not
necessarily be the calculated value [5 (c)]. The observations must rigorously satisfy
the equations

) ° — 0,

//•mT'Vja-t-iCo-l-Cr

or

The sum of the squares of the left side which contains the accidental errors is a
minimum for the most probable values of £, 77, and (n—l) values AFmn. The solution
gives these unknowns as functions of one of the AF, or if all the AF be introduced as
unknowns one of them must come out indeterminate. Instead of AF, I introduce the
corrections of the adopted times of exposure A<. For the first five photographs £, rj,
A£,, A^j, A<3 will be found as functions of A<4 and A£8, and for Photographs Vi., VI.,
and VII., f, 17, A<6 as functions of A<5 and A£7. Finally, all the time records can be
used in determining the corrections A.t. It will be seen that the uncertainty of the
exposure of Photograph VI. is not such a serious deficiency as might be expected at
first sight.

BLUE-VIOLKT LIGHT l\ THE SOLAR CORONA ON AUGUST 30, 1905. 323

To determine by the Method of Least Squares only ( and 17 would mean the
discarding of the condition (A, + .r )/(/!. + a;) = (h,\ + x)/(h'x+x) and must lead to
erroneous results.

I calculate £, ?/, and A< from equations which result from logarithmic differentiation
of the equation given above. Let tn = C+ Af.(n = 1 to 6), t: = 100-88-<,°-A<a+A<7
(see § 6), a = 1'05 + Aa, where tn°(n = 1 to 5) are assumed to equal the observed
values and C is arbitrarily chosen equal to 1 TOO (see § G). The values FM, are those
appearing in § 5 (<•), and they sufficiently approach their true values. I start from
xa = 140, yn = 4, the result of a first solution. The equations of condition are

n =
where

a= - Mod (1-F,.-1 ')(/*„,+

b = —a log Fmny cm = Mod (4a/m)~l, and similarly for suffix n,

but

e» = Mod(4a)~I (tt~l— t7~l) in equations m = 6, n = 7,

d = (4a)-I log Fw» for m — 5, n = 6 and 7/1 = 6, n = 7,

The weight p of an equation is calculated with ?•„ and rn as given in Table III.,
O'Ol being the error of an equation of unit weight. The calculated weights served
merely as a guide. The adopted weights appear in Table IV. The numerical
equations are : —

Photographs.
I. II.

I. III.

I. IV.

I. V.i.

V6. Via.

Vlft. VII.

n is entered in Table IV. The brackets indicate logarithms, —10 being omitted.

On account of the defects of Photograph VII. and the uncertainty of the exposure
of VI., I have solved the equations appertaining to Photographs I. to Va. separately
from those belonging to Photographs V6. to VII., and finally have discussed the
whole material.

(a) Photographs I. to Va. — These determine the intensity curve from distance 60
to 520. In accordance with the above, two of the A< are indeterminate. I choose At4

2 T 2

n = -(

n = -(9-239)(A1 + 140)-If + 0-0553i + (9-090)A/1 -(9-111) A/4

n = -(9-367)(A,

324 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

and A«6 and express the other unknowns as functions of them. The result of the

solution is

x = 140 +9-2 ±16 -22-2 A<< + 21'2 A/j

y= 4-0 + 0-71± 0-22- 4-68A/4 + 4-40A4

*, = 0-84-0-28± 0-08+ 3'01A*4- l'84A/5

/2 = 0-80-0-18± 0-06+ 2-28AA,- 1'20A*5

/8 = 0-78-0-01± 0-04+ 1-66&4- 0'65A/5

/4 = 0-80 + 1 A<4

<5 = 0-85 1 A<5

The errors are mean errors. The mean error of an equation of unit weight is
0'014, as compared with the adopted value O'OIO.

So far the time records have not been used (except in the calculations of the
differential quotients, which is merely a matter of convenience). I determine A£4 and
A?s from all the time records, introducing the condition that the values of t differ
from a mean value £0 by accidental errors v. The equations are

/„-» = 0-56 + 3-01 (A*4-A<S) +
t9-v= 0-62 + 2-28 +1-08

to-v = 0-77 + 1-66 +1-01

<o-f = 0-80+1-00 +1-00

/o-v = 0-85 +1-00

The result is A«4-A?5 = +0-096-0'05 A«8. The equations do not determine A<B
with any degree of accuracy. The unknowns then become

x= 147 ±16 +2 A/j

y = 4-26+ 0-22-0-05A4

i, = 0-85+ 0-08 + 0-98A4

/2 = 0-84 ± 0-06+1 -07 A<5

/»= 0-93+ 0-04 + 0-93A/5

<4 = 0-90 +0-95 A/.-,

tb = 0-85 +1-00 A/.-,

The value of A£8 is irrelevant for our purpose, it cannot be more than a fraction of
ft second and such a value changes x and y only by a small fraction of its error.

We may change y by a small quantity 77 of the order of its error and x by
a corresponding quantity £ without altering appreciably the residuals. The equations
give £ = (1'880) r). I assume y = 4 and throw its error on *. The result is

(A) z=127±23, y=4-00.

(6) Photographs Vb., VI. and VII. — The photographs furnish material for the
intensity-curve from h = 110 to about 1700 (17 solar diameter).

I'.LUE-VIOLET LIGHT IX TMK SOLAR CORONA ON AUGUST 30, 1005. 325

Again two of the Al remain indeterminate. I take A£7 = 0, which is permi&sible,
as any reasonable error has a small effect on x and y, see (c), and express the
unknowns as functions of AJ&. The result is

*- 140 -23 ±46 +1-4 A/s
y- 4-00- 0-29± 0-32-t-05A/4
/„ = 11-0 + 1-37 + 0-60 + 5-45A/4

The error of an equation of unit weight is 0'021.

The time records (except t^+t. = 100'88) have not yet been used. Afs can be
determined from the last equation.

The upper limit of J» [see § 6 (a)] is 11 -84-0-1 = 117 and it gives Af5 = -0'12
and the lower limit of tt is H'84 — I'O = 10'8, which gives A/5 = —0*29, both with an
error of O'l 1. The large value of A*6 belonging to the lower limit of tt is out of the
question, because the pendulum of the contact clock could not possibly have been
placed so much out of beat. Nevertheless, I maintain lx>th values,

/«- 11-7 /„ = 10-8

z = 117 ±46 z = 117 ±46

y - 3-84± 0-34 y= 4-01 ± 0-34

/i = 0-73 /j = 0-56

I again reduce x to y = 4'0. The normal equation gives £ = (2' 107) rj, hence

(B) /„ - n-7 /„ = 10-8

£ =137 ±63 t = 116 ±63

y = 4-00 y = 4-00

^ = 0-73 /a = 0-56

The second result is not possible, as already mentioned.

(c) All the Photograph I. to VII. — In this solution I have included the unknown
Aa in order to see what effect the error of a h»s on x and y. Five corrections AF
can be found or five of the A£, leaving two, say AJ5 and A*7, besides Aa, indeterminate.
The result is :—

t = 140 -2 ±19 -7 A/i + 0-5 -V7 - 20 \*
y= 4-00-0-15± 0-14-1-1 A/5 + 0 • 009 A/T - 4 Aa
^ /, = 0-84 + 0-21± 0-08+1-68A/5-0-006A/7+ 2-5 Aa
/2 = 0-80 + 0-19± 0-06 +1-46 A/& -0-004 A/7 + 2'0 Aa
/, = 0-78 + 0-27± 0-05+ 1-30 A/4-0-003 A/7+ 1-4 Aa
<4 = 0-80 + 0-17± 0-04 + l-16A/s-0-002A/7+ 0-8 Aa
<» - 0-85 +1 A/5

11-00+1-17+ 0-37 + 5-5 A/i-0-063A/;+ 0'25Aa
/? = 100-88-^ +1 A/7

326 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

The mean error of an equation of unit weight is found = 0'017 as compared with
the adopted value of O'OIO. Any possible error A£7 cannot alter the value of the
variables by more than a small fraction of their errors, and a was deduced from
experiments with an error of ±0'01. So far the time records have not been used;
I determine A<5 as under (a) from the recorded times tl to t&, neglecting A?7 and A«.
The result is A/r, = — 0'14±0'06. I substitute this value and calculate the errors on
the supposition that Af7 = ±l*'0, Aa = ±0'02, which certainly exceed the true
errors. The result is

re=139±19, y = 4'00±0-17,

tt becomes 1T40 — 0'50, which lies between the limits derived for te in § 6 (a). In
this solution no use has been made of the time of exposure assigned to the sixth
photograph.

The error of y may be combined with that of x (see above).

(C) o:=139±23, y = 4 '00.

The results (A), (B), (C) agree very well ; the good agreement of (A) and (B),
which rest on different material, is remarkable. Considering that all the material
contributed to (C), I might adopt it as final. I change x by a unit to round off the
figure. Hence

(D) / = <?(/; + 140 ±23)-*.

h is counted from the sun's limb in unit of 10~3 solar diameter and log c = 12'228
expresses the intensity in unit of the intensity of the corona at h = 1000. The
residuals left by (D) and calculated with the corrected values of F appear in
Table IV. under heading (v). I employ them to derive the errors of the distances.
I divide the residuals in each column in three groups and regard the mean of the
residuals in each group as the error of log(/iB+140)-log(/«m+140), where hn and hm
are the mean distances in each group. The errors of measurement will be about the
same on the first two photographs and they can therefore be calculated. The
calculations of the errors of hn is sufficiently evident. The result is :—

h. 100. 200. 400. 600. 800. 1000. 1200. 1400.

Errors. .1-4 5 15 27 47 70 100 130

These errors are almost twice as great as those in Table III. The excess must,
I think, be mainly set down to systematic errors of measurement which are different
for the several positives.

Fig. 7 shows a comparison of the intensity curve with the observations.
As the observations do not give absolute intensities, but the ratio of the
intensities at two correlative distances, I adopt at distances hlt AM, and /<«, the
intensities as calculated from the formula (D) and calculate the intensities at the
correlative distances ha, /i3, A4, /jta, h^, hlt from the latter and the known ratios F.

BLUE- VIOLET LIGHT IN TIIK SOLAR CORONA ON AUGUST 30, 1905. 327

The differences between these intensities and the tabular intensities are the out-
standing errors. The ratios of the intensities are calculated with 1 1 '40 seconds for
the sixth exposure and the recorded values of the times of exposure for the other

too

•H

I

1500 Scale for h
i

Intensity Curve.

Log i M-

-M-

Correjpondmq points.
N? of Photograph.

• o O O O «• O " £J
I I • ff % to «. VLH

2-0-

-1-5-

-1-0-

-03-

.-

TT

-W-

a

-1-0 •

Fig. 7.

328 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

exposures. The outstanding errors thus contain, apart from the errors of h, all the
systematic errors arising from erroneous records of times of exposure. The points
belonging to Photographs I., Vb., ~VIb. which ai-e placed on the curve are shown by
dots, while the observed correlative points are marked by circles. The systematic
errors are clearly reflected in these points.

§ 8. Question ivhether or not the Formula (D), § 7, hold* good at any

Position-angle ?

Table I. gives at intervals of 15° of position-angle the amount 8/1 by which the
distance from the sun's limb of an equal-intensity curve exceeds the mean distance of
that curve. Since these quantities were obtained from measurements on Photographs
I., VIII., and IX., which have a uniform background all round, systematic errors of
measurement will be eliminated in the differences, and their errors are more com-
parable to the errors given in Table III. than to those derived in last section. At a
certain point of an equal- intensity curve, which is h + oh from the sun's limb, the
intensity is expressed by formula (D), in which h designates the mean distance of
the intensity-curve. The intensity at the point may also be expressed by
(c+Sc)(A + 8/i + 140)~4, and if c + Sc be a constant, i.e. SA/A + 140 = 8c/4c = constant
at a series of points, the formula will hold good for these points. The values
a = 1008/J/A + 140 are entered in Table II. The value of the constant c + Sc, which
gives at /i + S/t a value of the intensity equal to that given by formula (D) involving
c and the mean distance h, is given by 8 log c = 0'017a, (0'017 = 4 mod/100). Accord-
ing to Table II. the quantities a differ for the same position-angle, and they vary
systematically with the distance. I adopt for the same position-angle the same
constant at all distances, and determine it by 8 log c = 0'017a0, and hence the
logarithm of the calculated intensity at h + 8h will differ by (a0— a) 0'017 from that
calculated by formula (D). I choose for a0 the mean of the values a belonging to
the same position-angle, and find that au— a lies between 0 and 4 for 90 per cent, of
the number of points and, therefore, the difference of the intensities is from 0 to 17
(log T17 = 0'068) per cent, of the intensity. An error of 17 per cent, in the intensity
is equivalent to an error in distance h of 9 at h = 100, 21 at h = 400, 44 at h = 1000.
The errors of h belonging to formula i = (c + Sc) (/i + 8A+140)~4 are those thus
derived combined with the errors given at the end of § 7. The residuals in h left
by such an intensity curve would, therefore, be in excess of the errors of the observed
values of h.

. At some position-angles 8c/c changes little with the distance from the sun and
therefore the formula represents the observations satisfactorily, and in some regions
the representation would be improved if points lying on a curve be considered
together. Whether these curves agree with the coursa of the streamers or not I
have not investigated.

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 329

§ 9. On the Number of Particles and Intensity of Light per Unit Volume of the

' Corona.

I shall explain that this problem can be solved on the following assumptions :—

(1) The luminosity of the corona is caused by particles, which are heated to
incandescence by solar radiation, and which scatter sunlight.

(2) The number N (r) of particles per unit volume is a function of the distance r
from the sun's centre.

(3) The apparent intensity is a known function of?- [see formula (D), § 7].

(4) The ratio g(r) of polarised and total light has been observed and represented
as a function of r.

(5) The intensity of light, T (r), of a particle heated by s«lar radiation is correctly
determined by STEFAN'S and the Wien-Planck formulae.*

With reference to (5) I have calculated the temperatures of particles at distances
h = 50, 100, 200, 300, 400, 600, ..., 1600 from STEFAN'S formula (absolute tempera-
ture of the sun = 6000), and the intensities for wave-lengths 3000 to 5000. I find
that their integral intensity T(r) appertaining to blue-violet light is very nearly
inversely proportional to the sixth power of r, the average error of the intensities
between /» = 50 and 1200 being only 7 per cent, of the intensity.

I adopt the following notation :—

C = centre of sun, P = position of scattering particle, r = its distance PC (in unit
of the sun's radius), 6 = £TJ— angle CP Earth, P (r) cos* 6 = light polarised by a
particle at P in direction 0, S(r) — P(r) cos* 6 = total light scattered by a particle at,
P in direction 6, F (r) = N (r) [T (r) +S (r)], f(r) = N (/•) P (r).

The functions are

->), P(r) = c, (r"-r"),t
T(r) = Clr-.

Let us find by integration the total light emitted by a channel of unit section which
runs in the direction towards the earth. I designate by p(= rcosB = h + 500/500)
the shortest distance of this channel from C and introduce

" sec 6 = 360 (A + 500)-'.

The element of volume at P = r sec 6d6 = £&g<7

Unit volume at P sends light F (r) -f(r) cos* 0 = F(gcos6) -f(g cos 6) cotf$.

The total light sent by all the particles in the channel towards the earth equals

(2)

* Sci- AKKIIIAM-.S, 'Lick Olwi-rvutory Bulletin,' No. 58.

t See Dr. SCHUSTER, " On the Polarisation of the Solar Corona," ' M. N.,' voL 40, p. 38 (6).
VOL. CX^VII. — A. 2 U

330 PROFESSOR L. BECKER ON THE DISTRIBUTION OP

The left side can be developed into a power series of g

(3) (1 -<?)- = i P^,
therefore F and f must also be power series of g :

(4) , .

(5) f(p)=f(g) = C'<fZnng*> where

Let

dn = f2 cos" e do.

Jo

,^ , 1.3...2a-l7r , 2.4... 2a

rf2a = 2^r

These substituted in (2) give

(7)
or

The second integral in (2) gives the polarised light, while the left side equals the
total light. Their ratio was designated by q (p) ; hence

Provided q (p) be observed as a function of p, i.e. of g, RB can be calculated by (II),
and Q by (1), i.e. F (p) and f(p) become known functions. Their values are, if
C" = C' (360/500)5,

FO») = NG>)ETG»)+8G>)]-

(III)

f(p) = K(p)p(p)

Substitute T, S, and P and find N (p) and c,/c2. The problem can therefore be
solved if q (p) were known. I am unable to say whether the measurements of the
polarised light made at the last eclipse suffice to determine this function.

With reference to (I), 2Pn/3e?ll+3 very nearly equal the coefficients of a binomial
series, and it is not difficult to prove that

The exponent y = 4, of formula (D), is derived from the observations with an error
of 0*3 (assuming x = 140 to be correct), hence the errors of the exponents in the

BLUE-VIOLET LIGHT IN THE SOLAR CORONA ON AUGUST 30, 1905. 331

above inequality are about ten times as much as the range of the exponents, and we
may write

(I), (4), and (5) give then, if C"' = l'5C",

-f(p) = CV8 l - -

which stands for the light radiated and scattered at right angles to the radial
direction by the particles in unit volume at distance p from the sun's centre.
Considering that the second term is only a fraction of the polarised light and the
latter a fraction of the total light, F(p) —f(p) nearly equals the first term. If there
were no light scattered by the particles but only radiated, the number of particles
per unit volume, N (p), would, by (1), be proportional to p(l— 0'72p~l)~4 5. This
result differs from that derived by ARRHENIUS, who based his calculations on
T (p) = constant.

10. Plea for Repetition of such Observations as contained in this Paper and for
Observations of the Light Polarised at Various Distances.

(a) For wave-lengths 0'3 to 0'5 the radiation of a particle at h = 50 is 355 times
as great as that at h = 1000, while for wave-lengths 0'55 to 0'65 this ratio is only 70.
Blue-violet radiation is almost inversely proportional to the sixth power of the
distance of the particle from the sun's centre (see § 9, 5), and for red-yellow radiation
the power is only 4'3.

Hence if in addition to photographs on ordinary plates a series of photographs be
taken with a colour screen on a plate sensitized for red-yellow rays another formula
would be found which should lead (see last section) to the same number of particles
per unit volume as that belonging to blue-violet radiation. Two such series of
photographs, together with observations of the light polarised at various distances,
would thus decide the debated question whether the luminosity is actually caused by
minute particles which are heated to luminescence by solar radiation and which
scatter sunlight.

(6) Though it is a fact that the brightness of the corona undergoes changes, we are
ignorant whether the intensity of the corona at a certain distance in terms of that at
unit distance is a constant or not. Inferences might be drawn from data such as
contained in this paper and belonging to a series of eclipses which would advance our
knowledge of the constitution of the corona and give us some idea of the causes which
produce it. It is, of course, necessary that the plates have on all occasions the same
relative sensitiveness in the different regions of the spectrum. (I employed Imperial
special rapid plates.)

2 u 2

332 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

I cannot finish this paper without expressing my indebtedness to the University
Court of Glasgow for a grant of £100 towards the expenses of the expedition ; to my
companion, Mr. JOHN FRANKLIN ADAMS, who presented half of this sum to the
Court and superintended the arrangements for the transport of the instruments ; to
Mr. ANDREW CROOKSTON, Glasgow, for his hospitality at his comfortable house at
Kalaa and the help the employees of his firm extended to the expedition en route ; to
the Council of the Royal Dublin Society for the loan of a siderostat, and to my
companion, Mr. HENRY A. MAVOR, M.Inst.C.E., Glasgow, who, in the capacity of
physician, engineer, and adviser, took upon himself much of that work which is not
mentioned, but is so important to the success of an expedition.

APPENDIX I*

Diffraction due to the Screens.

For the first four exposures, each of about a second, the aperture of the lens is
reduced by a perforated screen which has thirteen equal circular openings. The
arrangement of these openings will be seen in fig. 1 : there are six holes in the
corners of a regular hexagon, one in the centre, and six others are equidistant from
each two of them. The diffraction pattern of a star does not consist, as might be
thought, of a series of detached images which lie on lines intersecting in a centre,
but, as photographs of a Lyrse have proved, shows, apart from a central region,
luminous rings at the same distances at which one opening produces them. On the
photograph of a Lyrae rings are visible as far as 5n (linear value of TT = 9) for the first
screen, and on the eclipse photographs the prominences have certainly made no
impression beyond lOn-. Faint though the intensity of the rings be, it requires
investigation whether a distant ring belonging to a point of the corona near the sun
has an intensity comparable to that of a distant point on whose image the ring is
superposed, or rather whether all the diffracted light together is not a negligible
quantity. I shall show that it is small. The result would have been different if the
exposures had been longer and more distant parts of the corona had been photographed
with the screens.

Let P be a point in the focal plane and C the position of its central image.
I introduce a rectangular system of co-ordinates XY in the plane of the screen, origin
in centre of the central opening, and X-axis parallel to CP.

Let there be only two holes which lie diametrically opposite and whose centres
have the x-co-ordinates ±x, then the state of oscillation at P is given by

o

kirp3 - Jj (u) 2 cos (rx) sin a,

w

* Postscript, added at the request of one of the Referees. The photographic experiments were
subsequent to and confirmatory of the mathematical analysis.

BLUE-VIOLET LIGHT IN TIIK SOLAR CORONA ON AUGUST 30, 1905. 333
where p designates the radius of the opening, 0 the angular distance of CP at the

o_

centre of the object-glass, \ the wave-length, r = — sin 0, u = rp, and J, (u) BESHEL'H

A,

function of order 1. Let there be thirteen holes arranged as defined above and the
distance between each two be equal to a, and let a diagonal of the hexagon and the
ar-axis enclose angle <£, then the state of oscillation at P is given by

sin a UW- J, (n)\ 1 + 2 cos (ra cos <£) + 2 cos (ra cos(£ + </>)) + 2 cos (ra cos(^ -<j>\\
I it \_ \ w // w //

+ 2coB(ra^/3siji<f>) + 2coe(ra^/3co6\^ + ^U + 2 cos ( ray/3 cos fe -<f>\] I.

The intensity at P is the square of the coefficient of sin a. The position of point P
is determined with reference to C by its linear distance £ =/sin 0 (/= focal-length)
and its position-angle <^> counted from a line parallel to one of the diagonals of the
hexagon of the screen. For same values of £ the intensity is the same for ±<f> and it
is periodical with reference to <f>, with a period of ir/3. Hence the intensity can be
developed into a cosine-series progressing by multiples of 6$.

To find the quantity Q of light falling on a ring round C limited by radii £, and £,,
I multiply the intensity by the element £ ^£ f^0 of the area in the focal plane and
integrate from <£ = 0 to 2n- and from £, to £»• The integration with reference to <f>
can be carried out. The result is, if u be introduced instead of £,

u , a
'

Q.,, = V

+ Jfc1 (f\)' (irp>) 1 2 P {J> ("))" du [8 J0 (u') + 5 J0 (2«') + 2 J0 (3u')

• u, U

+ 6J0 ( ysV) + J0 (2 v/3V) + 4J,

J0 designates BESSEL'S function of order zero. I transform the second integral.
The values of J0 and J, are with sufficient accuracy for values of u larger than v,

- 4=«° (*+?)• (J>(^))S = ~ - C1-81" 2j:-)-

if \/x

The terms in [ ] have, for the first screen, respectively the periods 70°, 35°, 23°,
40°, 20°, 26°, and owing to these short periodic terms the quantity to be integrated
changes sign at small intervals of «. To a given value of uit say, = nir, a limit u,
near (n+\)jr can be found whicli makes the second integral zero. I have
convinced myself by mechanical quadrature that this deduction is correct even for
u, = 0, w, = TT, ..., K! = STT, Uy = 4ir. For our purpose it is unnecessary to take the
second part into account. Therefore, if all the light falling on the ring be considered

334 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

together the distribution is almost exactly the same as if all the light had passed
through only one of the openings ; and in accordance with the above, if the ring be
divided into 12 parts by 6 diameters, beginning at <£ = 0, each segment contains a
twelfth of the light falling on the whole ring.

The quantity of light falling on such a twelfth of a zone between u = mr and
(»+I)TT is given by

Qn = CtVlV/., where /, = 2 f" "

JIMT

U

i designates the intensity outside the object-glaas.

I next consider the light Q falling on unit area at a point C of the image of a
luminous area. Draw circles, radii mr, round C, and divide each ring by 6 diameters
into 12 equal parts. Project from centre of object-glass this system of circles and
lines on the luminous area. Let the intensity inm of the source be constant within an
area of the source corresponding to part m of ring n ; then, if there were x units in
one of these parts, each unit would send QB (for i = inm) divided by x to unit at C,
i.e. the x units send QB to unit at C. The total quantity of light falling on unit at
C is therefore given by

Q =

or

Q =
where

At = i

i

because 2 fn = 1. Parts 1 and 7, 2 and 8, &c., lie diametrically opposite with

0 *

reference to C. Hence the quantity of light at C is not changed by diffraction if the
source be everywhere equally intense, or if the intensity uniformly increase along the
lines drawn through C. In the case of the corona the second condition is very nearly
satisfied in the neighbourhood of a point, and thus the most luminous diffraction
rings hardly affect the quantity of light at C, there being almost as much light lost
as gained.

Let At be known for each of the four screens (i.e. p), and at each distance h.
Equal blackness was observed on two photographs exposed equally long with screens
a and 6, at two points ha and hh ; hence Qa = Q6, and

h
or

,b see § 5 (r)].

BLUE-VIOLET LIGHT IN THE SOLAK CORONA ON AUGUST 30, 1905. 335

The distances /t + A/i at which the corona lias the intensities t'+Ai are correlative
distances on corona [compare § 5 (d) and (/)], where A/» = + AI-JT.

The intensity formula for the corona ought to have been derived from the observed
values of h corrected by AA.

As to the calculation of AA, I obtained the intensities of the corona from a diagram.
I drew six lines through C at intervals of 30° and marked off points at distances
(n + $)ir from C. I assume that the intensity of the corona belonging to a point thus
marked equals the mean intensity at all the points lying within a ring limited by
circles HIT and (n+l)ir and up to 15° from it. The intensity at the points was read
off the diagram and multiplied by f,/l2. The linear value of ir is, in unit of 10~3
solar diameter, 9'1 for screen 1, 6'5 for screen 2, 47 for screen 3, 3'2 for screen 4,
0'55 for full aperture. For screens 3 and 4 several rings were treated together. In
some directions the calculation had to extend as far as ring 80n. I calculated f by
the following formulae

for large n,

"

Jffl(n7r)=

., 1 \
8nir/

4uir.

For small values of u I interpolated the value of the integral from the table given
in MUELLER'S ' Photometric der Gestirne,' p. 166.
The result of the calculation is —

Screen . .

1

2

3

4

h
AA . . . .
» . . . .

65 110 160
+ 0-7 -0-4 -0-9
1-2 1-7 3-5

110 210
-0-3 -1-1
1-7 5

160 270
-0-5 -1-0
3-5 8

210 350
-0-5 -1-5
5 12

The systematic errors AA due to diffraction, and still more their functions t» [see
§ 5 (/)]> are ao small compared with the accidental errors n of measurement, as
calculated from the residuals, that they can be neglected, and hence formula (D), § 7,
gives the relative intensities of the corona.

APPENDIX II.

Comparison of Corona and Moon.

The results contained in this section are not to be considered as an attempt to
standardise my eclipse plates, but they originated in a desire to give future observers
some ideas of the intensities with which they have to deal.

336 PROFESSOR L. BECKER ON THE DISTRIBUTION OF

After my return to the Observatory 1 photographed the moon on three nights
with the eclipse apparatus and approximately at the same zenith distance the sun
had at the eclipse. The atmosphere was exceptionally transparent for Glasgow on
the first and third nights. I used plates returned from Kalaa and I developed them
in the same way and at the same temperature as the eclipse photographs.

(a) Brightness of Corona. — The plates show, just as the eclipse photographs, a
background due to diffused light. I compared the intensity of the background
of the lunar photographs with those on the eclipse negatives, picking out those
exposures which showed the same density of background in both. The durations of
exposure give then the ratio of the light of the sky when illuminated by the corona
and that when illuminated by the moon, and this equals the ratio of the total light
emitted by the corona and the moon, provided the diffused light at the eclipse is
exclusively due to the corona and the relative intensities of the two spectra are the
same. Assuming ZOLLNER'S observations of the luminosity of lunar phases in terms
of that of full moon, I find from 17 comparisons that the total light of the corona
equals seven full moons. The comparison belongs to blue- violet rays.

(6) The Intensity of the Corona in Terms of Lunar Intensity. — Let ia be the
intensity of the region of the moon which lies north and south of Grimaldi and close
to the edge of the moon. I compared the blackness of this region on the photograph
with that of the corona on that photograph which was equally long exposed and
through the same aperture, and measured the distances of points of the corona at
which both showed the same degree of blackness. With the reduced distances I
calculate, by formula (D), i/c, which equals ijc. Instead of ia, which belongs to
phase angle a, I introduce ig, the intensity of the Grimaldi region at mean full moon,
and find log if/c equal to 2'543 from nine photographs on October 18, 2'578 from
three photographs on November 14, and 2'532 from 17 photographs on November 15.
The mean 2'551 belongs to h = 122, and at this distance from the sun's limb the
intensity of the corona equals that of the Grimaldi region at mean full moon.
Therefore the constant of formula (D) is Iqg c = 12'228 — 2'551 + log ig = 9'677 + log ig.
For want of suitable apparatus I am unable to measure ig in terms of the average
intensity of lull moon, but I am led to expect by integration of the intensity formula
and comparison with the total light of the corona (seven full moons) that ig is about 4.
it = 4 would make the intensity of the corona at a distance of 0'23 diameter equal to
that of full moon, a result which is quite at variance with that cited by LANGLEY.*

' The 1900 Solar Eclipse Expedition.'

RLUK- VIOLET I.ldlT IN THK SOLAU CORONA OX AICI sT 30, 1905. 337

TABLE I.— [S,-.- < I (c).j Ivjual-intensity Curves of the Corona.

MI-.HI distance

50

MO

400

,;,,,i

800

1000

PoHition-uiigle
from N. of sun.

Observed minus mean distances.
.Unit =0-001 solar diameter.

0

0

+ 6

+ 46

+ 78 +53

- 7

15

+ 4

+ 15

+ 43

+ 42 +62

+ 13

30

+ 6

+ 13

+ 13

+ 37 + 26

+ 4

45

+ 7

+ 11

+ 39

+ 43 +44

+ 59

60

+ 9

+ 24

+ 40 +48 +39

+ 19

75

+ 4

+ 37

+ 27 + 49

+ 15

- 54

90

+ 21

+ 11

- 2

- 9

+ 4

+ 9

105

+ 11

+ 11

-11

-22

-12

+ 11

120

+ 4

+ 6 -11

-28

-13

+ 6

135

+ 4

+ 4 0

-18

-11

+ 11

150

- 4

-11

- 6

- 9

-20

- 57

165

-24

-39

-50

-46

-50

- 37

180

-18

-24

-28

-38

-26

- 70

195

-11

-26

-32

-54

-34

- 82

•J10

-17

-48

-76

-74

-74

- 61

•_>25

- 4

-37

-68

-74

-94

-120

•-'40

- 4

- is

-50

- 57 - 67

+ 11

255

0

- 8

-41

-54

-45

- 28

i'70

0

+ 13

-15

K

+ 8

+ 35

285

0

+ 15

+ 11

+ 1

+ 24

+ 96

300

+ '6

+ 22

+ 57

+ 60

+ 63

+ 159

315

+ 4

+ 20

+ 50

+ 59

+ 52

+ 54

330

0

• 4

+ 11

+ 33

+ 28

+ 55

345

+ 2

+ 7

+ 52

+ 61

+ 28

- 26

TABLE II.— (See § 8.)

100 8/t

140'

Mr.m distance A

50

200

400

600

800

1000

Ponition-angle.

0

0

+ 2

+ 9

+ 11

+ 6

- 1

30

+ 3

+ 4

+ 2

+ 5

+ 3

0

60

+ 5

+ 7

+ 8

+ 7

+ 4

+ 2

90

+ 11

+ 3

0

1

0

+ 1

120

+ 2

+ 2

- 2

4

-1

+ 1

150

- 2

- 3

1

- 1

-2

5

180

- 9

• 7

- 5

- 6

-3

- 6

210

- 9

-14

-14

-10

-8

- 5

240

- 2

- 5

- 9

- 8

-7

+ 1

270

0

+ 4

- 3

- 4

+ 1-

+ "3

300

+ 3

+ 7

+ 11

+ 8

+ 7

+ 14

330

0

1

+ 2

+ 5

+ 3

+ 6

90 to L'70

- 2

- 4

- 6

- 6

-4

- 3

270 „ 90

+ 2

+ 4

+ 6

+ 6

+ 4

+ 3

VOL. CGVIL — A.

2 x

338

PROFESSOR L. BECKER ON THE DISTRIBUTION OF

TABLE III. — [See § 4 (c/).] Mean Corresponding Distances (h) from the Sun's Limb of

Points of the Corona at which the Photographs show Equal Blackness.

(Unit = O'OOl Solar Diameter.)

Photographs I. to V«.

I.

II.

III.

IV.

V«.

ft.

/'.

r-

li.

r. p.

/>.

r.

V-

A.

/•.

V-

h.

i
'"• ^

63

0-2

24

96

0-2

24

139

2-0'

24

187

2-4

24

263

4 10

69

1-1

24

102

1-6

24

154

1-3

24

209

1-9

24

283

3 11

72

1-5

24

111

1-1

24

161

1-7

24

220

1-7

21

309

4 11

74

1-5

24

109

1-8

24

170

2-4

24

224

2-8

21

324

6 11

74

1-5

24

111

2-4

24

169

1-7

24

228

2-8

21

326

7 11

76

1-1

24

111

1-5

24

165

1-3

24

220

1-3

21

309

4 11

85

•2-2

24

126

2-2

24

187

3-3

24

244

3-5

21

335

6 8

93

1-8

24

135

1-7

24

193

2-6

24

252

2-6

21

:!4:!

6 8

96

1-7

24

141

3-0

24

200

2-6

24

263

3-7

21

359

11 «

98

1-5

24

139

1-7

24

194

1-9

24

246

2-6

21

335

11 8

98

1-8

24

141

1-7

24

196

2-0

24

259

3-0

21

344

4 8

119

2-4

24

172

2-2

24

232

4-3

21

296

5-0

21

413

6 8

122

1-7

24

174

1-7

24

233

1-7

21

298

2-8

21

396

9 8

137

2-8

24

185

2-6

24

241

3-7

21

302

5

21

413

9 8

146

2-4

24

207

2-6

24

272

3-0

21

359

5

19

491

10 6

169

3-0

24

213

3-0

24

276

4-3

21

356

6

19

494

17 5

Photographs V6. and VIrt.

Photographs VK. and VII.

V6.

Via.

VIA.

VII.

//.

|

r.

V-

h.

f.

P-

//.

I'.

P-

//.

/•.

P-

*109

2

15

*333

10

11

*306

7

7

*572

6

10

159

5

15

443

5

10

365

8

7

683

17

9

187

5

11

450

4

10

424

11

4

831

18

9

*189

3

12

*469

5

11

428

7

4

774

17

8

207

4

9

548

23

10

*430

10

5

*804

11

7

209

4

8

517

8

9

463

20

4

826

19

3

214

6

9

567

5

7

504

13

4

902

15

3

232

4

7

533

14

7

*504

11

4

*1020

24

5

232

4

8

604

13

7

559

10

4

1026

44

4

*246

5

10

*611

7

9

609

35

4

1048

15

3

256

5

10

683

24

7

626

20

4

1204

14

4

261

3

6

574

9

7

*628

37

4

*1196

30

4

274

11

7

680

19

7

*667

1

2

*1295

15

.3

•280

4

8

*743

15

8

719

17

3

1028

35

3

304

3

4

696

9

6 809

28

4

1244

111

3

304

12

8

793

30

5 *963

26

2

*1556

48

3

315

9

7

739

9

7 *1094

1

*1778

60

:\

328

19

5

819 20

5

380

20

3

872

33

5

1

Measured on enlargements.

BLUK-V10LET LIGHT IN THK SOLAR CORONA ON AUGUST 30, 1905. 339

l •

l

<O ^" CO *— ^9 30 ^* CO *. PO ^^ t^ t* ^5 Si tO *"

+ +I + I++I l + l 1 l" + + + +

1

1

+ +I + I++I l-fl 1 I+ + + +

>

o ia ia e-i o TI >a o ?i •- >-

*

,

i

+ I++I + I + I I l + l 1 + 1 +

ar

T3

5
-o

~

ii + iiii + ili + iiili

i— f

"«• 1M

1 1

>

.

4

i

^^ CO l^ l^» 3i ^™ tf Tl CO w^ ^— F"— tC »i >O I1*

+ +i i ' + i + i++i+ + 'i +

•e

i

+ + +i i + + + + + + + + +I +

A,

ta 10 >a ia •* •*

PM

b

7"TTTTT0T7+0+5T5

•c

NJ

"c

+11111111+111+1+

*

0 L-. . - ,r.

hJ

"S

7"0T777TT++TT?T?

•c

1

"it

1 1 1 1 1 1 f t 1 1 I 1 1 ! 1 +

*

»O »fi

1

— • —

i

"3

F>H ~ ^

++I+ +111 III 1+

*

— «-—

1

i

liU

T
f

2 x 2

[ 341 ]

IX. On the Surface-Tension of Liquids Investigated by the Method of Jet

Vibration.*

liy P. O. PEDERSEN.

Communicated by Lord RAYLEIOH, O.M., Pres.ft.S.

Received June 11, — Read June 27, 1907.

[PLATES 2-4.]

CONTENTS.

Page

Introduction 342

Theory of the vibration of a jet about its cylindrical form of equilibrium 344

Calculation of the coefficient /*„ (x) 346

Calculation of the vibration of a jet 347

Preliminary investigations —

Arrangement for keeping the pressure constant 350

Determination of the cross-section of a jet 352

Production of the desired deviation from the cylindrical form of the jet 365

Determination of the wave-length 367

Investigations on the influence of the amplitude of vibration 371

Execution of observations ' 376

Various remarks 377

Remarks on the jet photographs 378

Result*—

Water 379

Toluol .384

Aniline 384

Aqueous solutions of ammonia 384

Solution of copper sulphate 386

Diluted sulphuric acid . 386

Aqueous ethyl alcohol . 386

Concluding remarks 387

* Abstr.ii tfil from a response to Det Kongl. Danske Videnskabernes Selskabs (The Royal Danish Scientific
Society's) probli-m in Physics for 1905; delivered October 30, 1906; awarded the Society's gold medal.
VOL. CCVII.— A 4-J1. 20.12.07

342 MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

INTRODUCTION.

AMONG the large number of methods available for the determination of the surface-
tension of liquids that proposed by Lord RAYLEIGH* stands out with great
fundamental advantages. The principle is as follows : — A jet of liquid issuing from a
not circular aperture is executing transverse vibrations about its cylindrical con-
figuration of equilibrium. Since the phase of vibration depends upon the time
elapsed, it is always the same at the same point in space, and thus the motion is
steady and the boundary of the jet a fixed surface showing stationary waves.

Measurements of the corresponding wave-length (X), the velocity (V), and cross-
section (A) of the jet, together with the density (p) of the liquid afford the necessary
constants for the calculation of the capillary-tension (T) according to Lord RAYLEIGH'S
theory of jet-vibration.

The method is free from every supposition respecting the angle of contact. This
advantage, however, it has in common with several other methods, especially the
following : —

The method of reflection proposed by R. Eorvost and also used by D. PEKAR| and

G. ZEMPLEN. §

The method of ripples || that has been used very much in recent times.
The method of maximum pressure of small air-bubbles proposed by M. CANTOR^

and further developed by R. FEUSTEL.*''

Another advantage of Lord RAYLEIGH'S method is that the surface in use is
continually renewed. In this manner the capillary-tension can be determined before
the surface is 7^5- second old. This circumstance is of very great importance, as the

* Lord RAYLEIGH, 'Roy. Soc. Proc.,' 29, p. 71, 1879 ('Papers I.,' p. 377).

t R. EOTVOS, ' WIED. Ann.,' 27, p. 448, 1886.

i PEKAR, 'Zeitschr. f. phys. Chem.,' 39, p. 433, 1902.

§ G. ZEMPLEN, 'Ann. d. Phys.,' 20, p. 783, 1906.

|| See for instance: L. MATTHIESEN, 'Pogg. Ann.,' 134, p. 107, 1868; 141, p. 375, 1870; ' WIED. Ann.,'
32, p. 626, 1887; 38, p. 118, 1889. A. ARENDT, 'Rep. d. Phys.,' 24, p. 318, 1888. KELVIN, 'Phil. Mag.,'
42, p. 368, 1871 ; ' Baltimore Lectures,' App. G., London, 1904. Lord RAYLEIGH, 'Phil. Mag.,' 16, p. 50,
1883 ('Papers II.,' p. 212); 'Phil. Mag.,' 30, p. 386, 1890 ('Papers III.,' p. 383). J. H. VINCENT, 'Phil.
Mag.,' 43, p. 411, 1897. N. E. DORSEY, 'Phil. Mag.,' 44, pp. 134, 369, 1897. J. A. CRAW, in A. GRAY,
'A Treatise on Physics,' vol. I., p. 659, London, 1901. L. GRUNMACH, 'Verb. d. Deutsch. phys. Ges.,' I.,
p. 13, 1889; 'Ber. d. Akad. d. Wiss.,' Berlin, p. 829, 1900, and p. 914, 1901 ; 'Ann. d. Phys.,' 3, p. 659,
1900; 4, p. 367, 1901; 6, p. 559, 1901; 7, p. 236, 1902; 9, p. 1261, 1902; 15, p. 401, 1904; 'Festschr.,'
L. BOLTZMANN, p. 460, 1904; 'Wiss. Abh. d. K. Norra.-Aich.-Komm.,' Heft III., p. 101, 1902.
KALAHNE, 'Ann. d. Phys.,' 7, p. 440, 1902. A. BRUMMER, 'Dissertation,' Berlin, 1902. K. LOEWENFELD,
'Dissertation,' Berlin, 1905.

f M. CANTOR, ' WIED. Ann.,' 47, p. 399, 1892 ; 'Ann. d, Phys.,' 7, p. 698. 1902.

** R. FEUSTEL, 'Ann. d. Phys.,' 16, p. 61, 1905.

INVESTIGATED BY THE METHOD OF JET VIBRATION. 343

main reason of the great discrepancies the different determinations of the surface-
tension show in relation to each other is certainly to be found in the variable
condition of the tested surface. These irregularities could arise from impurities, for
example, fat, oil, or similar substances, even the smallest portion of which is able to
produce a great alteration in the surface-tension. Thus Lord RAYLEIOH* has proved
that a film of oil not thicker than 2xlO~* mm. reduces the surface-tension of
water about 28 per cent., and the same author has latert proved that an oil film of
even 1 x 10~* mm. produces a noticeable reduction in the capillary-tension of water.
W. C. RONTGEN'S* experiments show that even an oil film of only 0*5 x 10~* mm. is
able to appreciably alter the condition of the surface. A. OBERBECK§ has been able
to detect the existence of a film of oil that was only 0'3 x 10~* mm. thick.

Apart from contamination, the surface can undergo different changes of a chemical
and physical nature. In this manner the fluid with which the surface is in contact
can cause a chemical change in it. If the liquid under examination is a mixture or a
solution the concentration at the surface will in many cases be different from that in
the interior.

From the above it is clear that the surface-tension of a liquid is, as a rule, not
constant, but varies with the time that has elapsed from the formation of the surface.
The value of the capillary-tension, immediately after the formation of the surface, I
propose to call the " initial value," while the value of the capillary-tension, when the
surface is sufficiently old, is called the " stationary value." Most of the methods for
the determination of the surface-tension give values differing from these two limiting
values, but it is just these two limiting values that have the greatest interest. Of
these the stationary value is of importance in many practical cases, but from a
theoretical standpoint the initial value is, without doubt, of the greater interest, as it
must stand in a more simple relation to the properties of the liquid than the
stationary value, which is dependent upon alien conditions.

With the other methods of measuring, attempts have also been made to work with
quite fresh surfaces. GBUNMACH, BKUMMER and LOEWENFELD have adopted a
method of obtaining pure liquid surfaces originally proposed by RONTGEN.|| The
principle of the method is, that the liquid is conducted from below through the neck
of a funnel over the upper horizontal edge of which the liquid flows. This method,
however, does not appear to be applicable to all cases (see GRUNMACH, ' Wiss. Abh.
d. K. Norm.-Aich.-Komm.,' Heft III., "Experiments with Mercury"), and has at
least two faults, firstly, that when the surface renewal takes place somewhat quickly,

* Lord l:\\ i nun, 'Roy. Soc. Proc.,' 48, p. 364 ('Papers III.,' p. 345); 'Phil. Mag.,' 30, p. 386, 1890
(' Papers III.,' p. 383).

t I.i.v-1 l; AYI.KK.H. ' Phil. Mag.,' 48, p. 321, 1899 ('Papers IV.,' p. 415).
J R.-NT..KA. ' WIED. Ann.,' 41, p. 321, 1890.
§ OBERHK K. • \\ n n. Ann.,' 49, p. 366, 1893.
|| RONTGEX, '\VIK.K Ann.,' 40, p. 152, 1892.

344 MR. P. 0. PEDERSEN ON THE SURFACE-TKNsioX OF LIQUIDS

it so easily causes inconvenient currents in the liquid ; secondly, the renewal is
slowest in the middle of the surface, just at the place which is the subject of the
measurement.

FEUSTEL* asserts that the method of maximum pressure of small air-bubbles also
gives the tension of a surface that is continually renewed. To this, however, may be
replied that the renewal takes place, and must take place, very slowly. If the air-
bubbles are produced quickly, the maximum pressure becomes dependent upon the
speed with which they are produced.

It will be seen that the surface renewal takes place by Lord RAYLEIGH'S method in
a much more effective manner than is possible with the other methods.

Notwithstanding the undoubted fundamental advantages of this method, it has
been used in very few cases, for besides Lord RAYLEiGHt it has only been applied by
F. PICCARD:}: and G. MEYER.§ Of these PICCARD has made use of the method for the
determination of the relative values of the surface-tension of ether, water, alcohol and
mercury, but his measurements were carried out with so great an amplitude of
vibration (see the plates of his paper, especially Plate VIII., figs. 14, 28 and 29, and
Plate X., Photographs 10 and 11) that his results are of very insignificant importance.
MEYER has only measured the relative values of the surface-tension of mercury under
various conditions.

The explanation of the little use that has been made of this method is to be found
in the great difficulties connected with adequate exact determination of the wave-
length and cross-section or velocity of the jet. It may be at once remarked here that
none of the methods previously used for the determination of these quantities can be
taken as satisfactory. It has, therefore, been of the first importance to work out
really good methods for the measurements of these quantities.

All the following measurements described here are carried out at ordinary laboratory
temperatures.

Even if this method is not so convenient in practice as some of the other methods,
that is no great drawback. What is needed in this field of investigation is not any
further accumulation of many different measurements, but some more reliable results.
Similar reasons have caused the method of ripples, which is just as complicated, to be
used a great deal of late.

Theory of the Vibration of a Jet about it* Cylindrical Form of Equilibrium.

\\. Before entering into the description of the experimental part of this work it is
necessary to set forth a few preliminary remarks on the theory of jet vibrations.

* FEUSTEL, loc. Hi.

t RAYLEIGH, 'Roy. Soc. Proc.,' 29, p. 71, 1879 ('Papers I.,' p. 377); 'Roy. Soc. Proc.,' 47, p. 281, 1890
(' Papers III.,' p. 341).

J F. PICCARD, 'Archives d. Sc. Phys. et Nat.,' (3), 24, p. 561, 1890 (Geneve).
§ MEYER, ' WIED. Ann.,' 66, p. 523, 1898.

INVKSTKlATKP I!V TIIK MKTIIOD OF JET VIBRATION. 345

It will be conveuient to set out together the meaning of the symliols employed:—

V = velocity of the jet (cm./sec.).
A = cross-section of tin- j»-t (cm.*).
p = density of liquid (gm./om.*).
T = surface-tension (dynr/i-Mi.).
Q = VA = discharge of the jet (cm.3/8ec.).

X. = 'iTTJk = wave-length corresponding to the vibration determined by for-
mula (I) (cm.).

Let us suppose that the jx>lar equation of the surface of the jet is

r = a + b, cos «<£ . cos kz ......... ( 1 ).

n is an integer greater than 1. The jet is here and in the sequel regarded as
horizontal and the plane </> = 0 is also horizontal. According to Lord RAYKEIQH'S*
theory the surface-tension is determined by

T_ y7T . _

"yFZJtt'SSRfia!}'*' x.« P

where

, tf,l\ - *S* l" (ak)
M"( ~a'P + n'-rakl'nak

Vibrations corresponding to different values of n in (1) will lie independent of each
other.

The development of Lord RAYLEIGH'S theory rests upon certain suppositions,
viz. : —

1. That the deviations from the circular-cylinder form are exceedingly small.

2. That the vibrations are executed without any loss of energy.

3. That the original velocity of the jet is the same over the whole cross-section.

4. That the surface-tension is constant.

Each of these hypotheses will now be viewed somewhat closer individually : —

1. This hypothesis is, in practice, impossible to carry out, as it is precisely on the
basis of the divergence from a cylindrical form that it is possible to determine the
\\ave-length, and the smaller the divergence the more difficult the determination
becomes. To reduce the uncertainty resulting from this, I have investigated a jet of
the same liquid partly with large, and partly with proportionally small deviation from

* KAYI.KIGII, 'Roy. Soc Proc.,' 29, p. 71, 1879 ('Papers 1.,' p. 377).
VOL. ccvn. — A. 2 Y

346 MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

the cylinder form, and I have similarly used a method for the measurement of the
wave-lengths that even permits of a really good determination for small divergences.
This matter is more fully considered later.

2. This hypothesis is also of great importance for the development of the theory,
but, on the other hand, not satisfactory in practice. The liquid has always some
viscosity even though, as in many cases, it is only small. It is, however, possible for
the most part to determine the influence of viscosity on the time of vibration, and
in this manner to correct the errors caused by it.

The calculation of this correction rests upon the following supposition, which will
be very nearly true as long as the viscosity is small :—

The harmonic vibration of the jet corresponding to the normal co-ordinate bH is
changed by the viscosity to a damped harmonic vibration.

Let the logarithmic decrement of the vibration be S ; we have then

where N, is the frequency of vibration with damping, N is the frequency without it.

For the determination of the surface-tension we have instead of (2) the following
equation

8M .<-).,

The experimental determination of 8 is described later.

3. The velocity of the thin jets investigated in this work will certainly be nearly
the same over the whole cross-section, and correspond to that calculated from the
cross-section and the discharge of the jet.

4. The surface-tension is in many cases dependent upon whether the surface
extends or contracts (compare, for example, the damping action of oil films on
waves) ; but with the fresh surfaces as used here the surface-tension is certainly very
nearly constant.

Calculation of the Coefficients p.n (x).

§ 2. The use of the formula [(2), § 1] for the determination of the surface-tension
demands the calculation of the coefficients pn(z) determined by the formula [(3),
§ 1], or

Here

In(x) = i-"Jn(ix),

where J, is the BESSEL'S function of the nth order.

INyESTIOATED BY THE METHOD OF JET VIBRATION. 347

Similarly r. (.r) = y**. In accordance with the theory of BESKKL'S functions

' './

we have

T' --JtT-i-T I' -- T -WT T i- ^" T — T -0 (9\

in l. + l. + i, * » — A«-i— - 1«. l.-u-r - 1» — l»-i - • • • \&)'

X X . X

Accordingly we have

M» (•**) = -5 ; — r • -T — ' — f = j * — r ' .... (3).

or +7i— 1 arl,-!— ?tl. or+n— 1 !„_,

x. -y-= — n

By use of the last formula (2) the values of I, and I3, Ac., can be calculated from
Io, Ii, and by substituting these in formula (3) we have ft, (x).

In order to facilitate the use of this method for the determination of surface-tension
I have calculated a table of the values of /^, (x) most commonly used. This table will
be found at the end of this paper, and contains the values of log,0 pm (f) for n = 2, 3,
4, and 6 and for x = O'OO to x = I'OO.

The details of the calculation of this table are given in my original paper.

Calculation of the Vibration of a Jet.

§ 3. In accordance with Lord RAYLEIOH'S theory the vibration of a jet can now be
determined when the velocity and original cross-section is known, although, as pre-
viously emphasised, the theory is only available for small deviations from the circular
form.

The circumference at the original cross-section is determined by

By help of FOURIER'S series this equation can be written as

MNB

r = <*„+ 2 bn.cos(n$ + f,) ........ (2),

when, if necessary, the value of «„ is changed so that ba vanishes, and the origin of
co-ordinates is changed so that 61 becomes zero.

Each term in (2) can be taken alone and the resulting vibration of the jet can lie
calculated as the sum of all the vibrations corresponding to the different values of n
in (2).

Thus it is only necessary to consider

r = a0+&,.cos(n^+e,) ......... (3).

The wave-length X, corresponding to the vioration (3) is, according to [(2), § 1],

where

C = />tfl.A*4.V.T-w (5)

is independent of n.

2 Y 2

348 MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

The determination of X, is easiest carried out in the following manner : —

= c • \« *> T - &c

In nearly all cases it will be sufficiently correct to take X,, = X"H.

If there is only one vibration (3) present, the equation for the surface of the jet

will accordingly be

r = nn + bn cos (>t<> + fn . cos 2irzXH.

If all the partial vibrations are taken in the same manner, we have, by the addition
of the results, the following equation for the surface of the jet

/• = aa+ 2 . &„. cos («<£ + eB) . cos(27T2/\B) (6).

As an example, we will here calculate the vibration of a jet the original cross-
section of which is an ellipse with the axis 2a and 2b = 2 (a— 8).
The polar equation of the ellipse is

ab a— 8

r =

v/a3 sin3 <£ + b3 cos3 <f> \/\—x cos2
where

(7),

-28-82

— ^ 5-

By expanding in series

„.
a cr

,COS2A^ -lj-4-1 i-^V + Il 1 3-S :r3+JL iiJ-iA.--',,.* . 1M .l_- » •»•.?.•_» yS. \

I- OUb -i<p.^a • a-^^a 2.4X ^39' 2.4.6<1// ^16 2.4.6.8^ ^256 2.4.6.8.10'*- T •••)

H>0«4^^ »^.S~» . « 1_-JL^5«3 , _7_ 1.8.5.74.15 1.8.5.7. 9 g , N
*9-U ' 2 . 4ar'1' 16 ' 2.4. e* ~T 3 2 ' 2.4.6.8X +64' 2.4.6.8.1Oa; + •••)

Umvafi^./1 1-JLL*1Jj. * 1-3.5.74, 45 1 . 3 . 5 . 7 . 9 5 , 5 5 1 . 8 . 5 . 7 . 0 . ll^g , \
°9 • ^ 3 2 ' 2 . 4 . 5* T"T¥' ITSTfTi* "t"512'2.4.e.8.101*' "*"512'2.4.6.8.10.12X T...;

Ur>/«i8/A/— 1 i-La_LAjJ~«J--A- 1-3.5.7. 9 ^ , 33 1.3.5.7. 9 . lie , \
0<P' Vl28 ' 2.4.6.81// """ase" 2.4.6.8.10'*' 1"1024' 2.4.6.8.10.12X T*««J!

+...} . . . :-. .... . '. y . v . I . . .-...-;; . (s).

By inserting .r = 0'4 or S/a = 0'2254 in formula (8) this formula is reduced to

r = 07746a(l-1318 + 0-1440 cos '2<£ + 0'0138 cos 4^

.) . . (9).

The further calculations are based upon the following constants : V = 240 cm. /sec.,
A = 0-066 cm3., a = 0'1647 cm., p = 1, and T = 73 dyne/cm. The wave-lengths
corresponding to n = 2, 4, and 6, calculated by the above given means, are

X3 = 3'932 cm. ; X4 = 1'227 cm. ; X« = 0'646 cm.

INVESTIGATED HY THE METHOD OF JET VIBRATION. 349

The equation of the jet Ijecomes then

r = 07746 . 0'1647 /Vl318 + 0'1440 cos 2<j> . cos J^L+0'0138 cos 4<£. cos -22L+ ...)

= 0-1444 + 0-01837 cos 2<£. cos

3*932

. cos 4<. cos

1 '

+ 0-0001913 cos 6<A. cos -

'

0'04G

(10),

from which the co-ordinates for every point of the surface of the jet can l>e calculated.
The profile line resulting from <£ = 0 in (10) has special interest. This line is shown
in fig. 1. In order better to judge the form of the curves the height is enlarged

10crr\.

Fig. 1.

fifty times in relation to the length. It can be seeu that the form of the curve is
mainly determined by the original vibration corresponding to n = 2, but that at the
same time also the other vibrations cause perceptible deviations.

In the measurements made by Lord RA.YLEIGH, PICCARD, and MEYER the wave-
length X, is determined as the length between two successive summits of the profile
line of the jet. It can be seen in fig. I, that this length can vary and deviate
somewhat from the wave-length X. In order to illustrate the size of these
deviations for the jet corresponding to (10), drawings of the summits of the profile
line are shown in fig. 2. By calculation it is found that r is maximum for
z = 0, 2 = 3-932-0-109 cm., z = 2. 3'932-0'171 cm., and z = 3. 3'932 + 0'211 cm.

As above stated, X, = 3 '932 cm.

The wave-lengths measured in this manner are

X0-' = 3-823 cm. ; X1'" = 3'870 cm. ; X11'"' = 4'314 cm.

The errors stated in per cent, of X3 are respectively — 2'6, — T6, and +97.

The error can be reduced by taking the mean value of several lengths. On the
other hand, the amplitudes of the supplementary vibrations have been greater in
proportion to the fundamental vibration with almost all the measurements up to now
than in the instance mentioned here.

Even with relative measurements as those made by MEYER* these reasons can

* MEYBR, for, eii.

350

MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

have weight, as the wave-lengths A,, X3, and X4 are dependent upon the surface-
tension, aud a slight variation, for example in X2/X4, can to some extent alter the
wave-lengths measured.

o
o

1cm

Fig. 2.

PRELIMINARY INVESTIGATIONS.
Arrangement for Keeping the Pressure Constant.

§ 4. Before I begin the description of the different measurements I will mention a
method for the production of a constant pressure by use of ordinary tap-water, as I
have used this means in almost all the preliminary investigations.

It has hitherto always been an inconvenience in experiments with jets that the
pressure varies continuously as the fhlid runs out. To avoid this variation it is not
advisable to use the simple method of renewing the quantity of discharge by a
corresponding inflow of fluid, as this arrangement produces disturbances in the fluid
mass, causing irregularities in the jet. All those who have worked with jets know
how great is the demand for rest in the reservoir, and how exceedingly sensitive the
jets are to external influence.

Lord KAYLEIGH* states : " The jet is exceedingly sensitive to disturbances in the
reservoir, and no arrangement hitherto tried for maintaining the level of the water
has l>een successful."

* Lord RAYI.KKJH, ' Roy. Soc. Proc.,' 29, p. 71, 1879. (' Papers I.,' p. 380.)

INVESTIGATED MY THE MITHOD OF .IKT VIl:i;\T!nN

351

After a number of CX|M-I iments 1 have come to the conclusion that the following
arrangement for this s]XM-i;il use is perfectly satisfactory:—

From the tap the water is conducted direct through a rubher tube to the spout of
the funnel T (fig. 3), which is fastened with sealing-wax to the neck of the bottle F.

To jet

Fig. 3.

This rests through a wooden frame R on a metal plate P, which is provided
with three adjustable screws S. The whole is borne on a bracket K, placed on an
outer wall. The bottle F is open above and is provided with an outlet from below as
shown in the figure. By help of the adjustable screws the upper edge of T is kept
horizontal.

The water coming from the supply pipe will run over the edge of the funnel in the
form of a thin layer, and the height of the surface of water in the funnel will only be
very slightly dependent upon the speed of the supply, so that the unavoidable
variations in the pressure of the supply pipe will practically have no influence.

The water in the funnel T is in connection with the water in the reservoir B
through a syphon made of glass tubes rt, rt, ra and rubber tubes </, and «/2; the
surface in B will keep the same height as the surface of water in T. B is provided
below with a tubulure that serves for the introduction of one branch of a T-tube, the
other two branches of which are provided with rubber tubes, the one serving the jet
apparatus, whilst the other is only used for filling or emptying the reservoir.

It follows from the above that when the quantity of water supplied to T from
the supply pipe is greater than that used in the jet apparatus, the surface of water

352 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

in B will keep itself practically constant, independently of the quantity used in the
jet apparatus. This is, of course, on the understanding that the diameter of the
syphon is sufficiently large.

Experiments have shown that with this arrangement, and for jets not exceeding
2 mm. in diameter, the water surface in B will vary at most 0'2 mm., and the
apparatus can stand and operate by itself day and night.

The above-mentioned arrangement is especially convenient for investigation of a jet
produced from ordinary tap-water, and was used in practically all the preliminary
investigations for judging the exactness and practicability of the methods of
measuring. As these investigations take, as a rule, a long time, it is very important
that the pressure be kept constant.

For other fluids that are available only in limited quantities this method cannot, of
course, be used. In these cases the author has, as a rule, employed the usual modus
operandi with decreasing pressure. This will be more fully explained under the
description of the experiments.

Determination of the Cross -section of the Jet.

§ 5. When using the present method for determination of the surface-tension it is
necessary besides the wave-length to know two of the three following quantities :
velocity of the jet, the sectional area of the jet, and the discharge. This last named
is easiest to determine with sufficient exactness, and will, therefore, in every case be
measured. The choice then remains between measuring the velocity of the jet or the
cross-sectional area, but before making this choice I will give a short summary of the
methods that are available at this moment to determine the velocity and sectional
area of a jet.

The velocity V can be determined by use of TORRICELLI'S formula V = ay/^H),
where H is the pressure, g the acceleration of gravity and a a coefficient. Many
experiments have been made to investigate the exactness of TORRICELLI'S formula.
The results of these investigations are, mainly : for fluids with little viscosity, with
not too high pressure, and, lastly, with holes the diameter of which exceeds 5 mm.,
the formula is practically correct, as the coefficient a is very nearly equal to 1 ; for
water a = 0-97 to 0'99. Special reference can be made here to TH. VAUTIER'S*
careful investigations on this subject.

With this in view, it was the author's original intention to determine the velocity
of the jet in this manner ; it, however, soon appeared that, just in the circumstances
which have especial interest in the present instance, the deviations from TORRICELLI'S
formula are very important. It is for this reason that, so as not to use too great a
quantity of fluid, it is necessary to use thin jets, for example with a diameter of

* TH. VAUTIER, ' Compt. Rend.,' 103, p. 372, 1886; ' Th^se pn's. a la Fac. de Science d. Paris,' 1888;
'Ami. China. Phys.,' (6), 15, p. 433, 1888; ' Journ. d. Phys.,' (2), 8, pp. 301, 396, 1889.

INVESTIGATED BY IMF. METHOD OF JET VIBRATION. ;15H

1 millimetre. Several inthienees result fn>m this which with a greater diameter of jet
have only secondary importance, hut lien- take a prominent part. I shall quite super-
ficially treat these influences here, as I hope later to have an opportunity to give a
more exhaustive account • >(' this suhject.

In Tonuii 1:1.1.1 's formula H indicates the pressure measured from the jet to the
fluid's .surface; but this in reality must he reduced by the pressure produced by the
surface-tension in the interior of the jet. Let </ cm. lie the diameter of the jet, p t In-
tensity, and T dyne/cm, the surface-tension, then will the pressure produced by the
surface-tension correspond to a head of liquid, the height of which, h cm., is

determined hv

2T

» ...» : ......... (I).

If we take, for example, for water d = O'l cm., p = 1, T = 73*5 dyne/cm., we have
li = circa 1*5 cm. As can be seen, it is a correction which is not quite infinitesimal,
though but little attention has been given to it. C. CHRISTIANSEN* is probably the
first who has commenced the investigation with special regard to the conditions under
discussion. Some experiments of M. IsARNt also confirm the alx>ve view. He
determined the time that elapsed for 141 cm:t. of fluid to run through a circular hole
of a diameter of 0'8 mm. with the pressure varying from 11 '8 cm. to 9'0 cm. and

for water 290 seconds,
„ alcohol 270 „

These two measurements will now be calculated with reference to the correction
mentioned before (l), it being supposed that the diameter of the jet in both instances
was d = 0'07 cm.

For water we take p = 1,1 = 73'5 dyne/cm. ; for alcohol we take p = 0'8, T = 22'0
dyne/cm.

In accordance with this the above correction will be for water h = 2'14 cm., for
alcohol h = 0'80 cm.

The effective pressure will according to this be

For water H = £ (v/ll-8-2'14 + v/9'0-2-14)' = 8 "20 cm.

„ alcohol H = | (v/H '8-0-80 + s/9-0-0'80)s = 9'53 cm.

The total discharge will be

0-07*

For water IT . - - • s/2'981 . 8-2 . 290 = 141 -3 cm3.
4

0-07*

„ alcohol ir . — — • v/2'981 . 9'53 . 270 = 141'8 cm3.

* C. CunisTiANSEX, 'Overs. \ "i.lonsk. Sclsk. Forh.,' p. 65, 1901 ; 'Ann. d. Phys.,' 5, p. 436, 1901.
t M. ISAIIX, ' Journ. d. 1'hys.' (1), 4, p. 167, 1875.
VOL. CCVII. — A. 2 Z

354 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OP LIQUIDS

Thus the difference shown in the time of outflow is wholly explained in this
manner.

ISARN himself explains the difference mentioned as originating from different
contractions, and calculates on the basis of such measurements the coefficient of
contraction. There can be no doubt, however, that this is incorrect, as the influence
of the surface-tension on the coefficient of contraction certainly is not great, as will be
shown later.

If this reduction of the velocity on account of the capillary-pressure in the jet were
the only deviation from TORRICELLI'S formula, it could be corrected and the velocity
accordingly calculated ; but as the diameter of the jet becomes smaller the value of a
is also reduced and this coefficient becomes to a great degree dependent upon the
nature of the edge of the hole, so that in every case it is necessary to determine the
value of a, or, in other words, determine the velocity of the jet in another manner.

Lord RAYLEIGH*, who used this method to determine the velocity of the jet, says
with regard to it : " The pressure at any moment of the outflow could be measured
by a water manometer read with a scale of millimeters. Some little uncertainty
necessarily attended the determination of the zero point ; it was usually taken to be
the reading of the scale at which the jet ceased to clear itself from the plate on the
running out of the water."

According to the above, this method can not be taken as a satisfactory solution of
the question.

Direct measurement of the velocity of the jet can be made in several manners.
TH. VAUTlERf added small drops of another fluid and determined the velocity of the
drops by taking photographs on a plate moving with a known speed. The method
seems to be good so long as the diameter of the jet is ' not too small (in VAUTIER'S
experiments the diameter of the hole was 576 mm.), but with small diameters the
method is useless on account of the risk of noticeable change both in the surface-
tension and in the coefficient a on account of the additional alien liquid.

Another and simpler method J is to determine the velocity by help of the geometric
form of the jet. This method can also give satisfactory results for thick jets, but for
thin ones it is of no value.

Besides the previously mentioned reduction of velocity on account of the capillary
pressure in the jet, the surface-tension produces other differences in the velocity.
The presence of the jet is inseparably connected with a continual production of new
fluid surface, and the requisite energy is essentially taken from the kinetic energy of
the jet as its horizontal velocity reduces. The loss of pressure, ht, corresponding to
this reduction is easily found to be ,m

h-f. (2).

dpg

* Lord RAYI.EIGH, 'Roy. Soc. Proc.,' 29, p. 71, 1879 ('Papers I,' p. 375).

t TH. VAUTIER, loc. tit.

} See WINKELMANN, ' Handbuch d. Phys.,' I., "Ausfluss uud Strahlbildung," F. AuERBACH, 1891.

INVESTIGATED l:V TIM. MKTIIOD OF JET VIBRATION. 355

This reduction is thus seen to be double that originating from the capillary-pressure
for a water JH with u diameter of 1 mm. Incomes A, = circa 3'0 cm.

This fact has also been the subject of but little attention, though A. DUPR£* has
undertaken some interesting experiments alxmt the height to which a jet can rise.

A closer examination of how the reduction of velocity corresponding to A, spreads
itself over the jet I am obliged to leave to another opportunity, only a particular
characteristic fact l)eing named here. If the pressure is reduced more and more so
that it comes near to the value ht + h, the jet can be observed to deviate more and
more from a paral>ola, the curvature of the jet just outside the hole becoming much
too great. If the pressure is reduced almost to Ai + A, the jet first runs a short
distance horizontally and then falls vertically down. If we determine the discharge
and cross-section of the jet, the velocity in the horizontal part of the jet can be
calculated, and it will be observed that this velocity is almost equal to that
corresponding to the pressure 1^. It is very difficult to maintain the jet in the above
position ; the slightest disturbance will cause the jet to cease.

It is also possible to determine the velocity of a jet by measuring the pressure it
produces by normal impact on a sufficiently large plane surface. Measurements in
this manner have been made, for example, by BoFF.t who worked with a jet with
diameter from 5 to 7 mm. This method seems to give quite reliable results and may
most probably be available even for much smaller jet thicknesses, but the difficulties
connected with its use will be, in consequence, considerably greater. The balance
used must, in such a case, be made very sensitive, and it will probably then be
difficult to keep the sensitiveness constant. In the use of this method there must
also be taken into consideration a correction resulting from the surface-tension. The
pressure measured must be reduced by the capillary-pressure in the jet multiplied by
the area of its cross-section, in other words, by

..=.
d 4 2

At the same time the surface-tension along the jet's circumference, or TTK/, must
be added to the pressure measured. The final correction will accordingly be 4- ^TrTrf.

The cross-section of a jet has hitherto, as a rule, been determined by direct
geometrical measurements, which ordinarily take place in such a manner that the
points of some micrometer screws are brought exactly to touch the surface of the jet.
In this manner a sufficient exactness can be reached, as a rule, with thick jets. The
condition is quite the reverse when the diameter of the jet is only about 1 millimetre.
In this case it will most probably be impossible to get even a moderately satisfactory
exactness. In this respect a great progress has been made by the elegant method

* A. DtiPRfc, ' Throne m&anique do la chaleur,' p. 376, Paris, 1869.
t BUFF, 'Poou. Ann.,' 137, p. 497, 1869.

2 Z 2

MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

proposed by K. PRYTZ* and based on the optical contact between a microscope and ;i
reflecting surface. It is possible in this manner to determine the diameter of a circular

B

20cm.

Fig. 4.

JH,

\

jet with great exactness, but with not circular jets the "method, unfortunately, becomes
impracticable.

* PRYTZ, 'Overs. Videnek. Selsk. Forh.,' p. 17, 1905: 'Ann. d. Phys.,' 10, p. 733, 1905.

IN\rsT|i:.\TKI» KV THK METHOD OF JET VIBRATION

357

§ 6. A.& none of the known methods for the determination of the velocity and cross-
section of thin jets are quite satisfactory, I have worked out a new method for the
determimition of the cross-sectional area of a jet.

b

Scm

Fig. 5.

Section

a-b.

c-

5cm..

Fig. 6.

The principle of it is the following : A definite length of a jet is taken and weighed,
and on the basis of the weight and the density of the liquid the sectional area of the
jet is calculated.

The taking of this definite length of the jet is accomplished by help of a "jet-
catcher," shown in figs. 4-7. It consists of two cylindrical vessels KI and K,, the

358

MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

ends of which are the steel plates l\ and P^ The edges of these steel plates, k, are
knife-formed and effect the cutting of the jet, as described later.

The "jet-catcher" itself is, as is shown in figs. 5-6, arranged on a pendulum C,
turning about the axis p-p. In order to vary the height of the pendulum the stands
A are furnished with a number of holes for the screws S, as shown in the figure.
The "jet-catcher" is fastened to the pendulum through the cross-piece t by means of
the screws * (see fig. 4). The exact position is secured by means of two cones i,
which fit in the corresponding holes in the cross-piece t.

5cm.

Fig. 7.

The axis of the pendulum is arranged perpendicularly over, and parallel to, the
horizontal jet at such a height that the position of the jet in relation to the "jet-
catcher" is about as shown in fig. 6, where S indicates the jet.

If the pendulum is caused to swing, the "jet-catcher," every time it passes the jet,
will cut out a portion of it. If the jet is perpendicular to the edges k, and parallel to
the plane through corresponding edges, the length of the piece that is cut out of the
jet each time is equal to the distance between the edges k. With a complete swing
(both forwards and backwards) the total length L of the portions of the jet cut off (see

fig. 7) is

L = L, + L2 . . . (1).

TNVESTICATKI) BY TFIK METHOD OF JKT VIBRATION. 359

If the jet is parallel to the plane through the edges, but makes an angle of 90°— <£
with these, then the total length of the portions cut oft' during a complete swing
will l>ecome

If the jet is at right angles to the edges, but makes the angle to with the plane
through the edges (see fig. 7), the conditions are a little more complicated. In
the figure the jet is drawn in two positions, S, and Sj. The one position, S,, gives a
picture of what takes place with the "jet-catcher" moving in one direction, S, gives
u corresponding picture of the movement in the opposite direction. V is the velocity
of the jet. With regard to the other symbols, reference is made to the figure.

We have

I/, = L, sec a +1/1, t/, = V/t'i . .r,, or, = LI tan &>,
also

L', = L^sec oi + V/u,. tan o>) ........ (3).

In the same manner we get

L', = L,(seca>-V/rs.tana») ........ (4).

By the addition of (3) and (4) we get

L = L', + L',= (LI + L,)[8ecw + £Vtan<o(l/r1-l/r!l)] . . . . (5);

as the last term is so small that we can take L, = L, without any appreciable
error.

If v, = tfe

L = (L, + Lj,).seca> . ' ........ (6).

In practice vl and vt will have almost the same value, although the velocity will
naturally be somewhat smaller each time the "jet-catcher" passes the jet. To
investigate the influence of this difference in velocity we take v3 = 0'9t',, V = r,.

The equation (5) then becomes

L = (Ll + Ls).(sec&>-0>0555tana>) ....... (7).

By t>, = 0'9r, and V = t'a, equation (5) becomes

L = (L, + Lg). (sec 01 + 0-0555 tan co) ....... (8).

To judge the influence of the angle o» the Table I. is available, which also contains
the corresponding values for rs = 0'95r, and r, = 0'95v^

In practice the ratio between v, and va will still more approach to 1. As can be
seen, no especially great exactness is required in the adjustment of the "jet-catcher"
relatively to the jet.

360 MR. P. O. PEDERSKN ON THE SURFACE-TENSION OF LIQUIDS

TABLE I.

w =

r.

2e.

3°.

4°.

5°.

ra = 0-90i'i

sec to - 0 • 055 tan <u

0-99918

0-99867

0-99846

0-99856

0-99896

r, = 0-90fo

sec <o + 0 • 055 tan <u

1-00112

1*00266

1-00428

1-00632

1-00868

«>2 = 0-95»-i

secw -0-026 tan w

0-999G9

0-99969

0-99999

1-00060

1-00152

r, = 0-95r2

sec w + 0 • 026 tan <o

1-00061

1-00153

1-00275

1-00428

1-00612

In these evolutions it is supposed that /, = 12 (see fig. 7), a condition easy to satisfy
with great exactness.

It can easily be seen that all the foregoing continues to be true in the main, even if
the velocity of the jet is not the same over the whole cross-section. With use of the
method on jets that are not cylindrical there are some complications. Reference will
only be made here to the jets investigated in this paper, the equation of which is
r =• a + b cos n<f> . cos kz.

When n is an even number, the oblique sections produced by the edges k can,
without appreciably altering the volume of -the piece cut out, be replaced by the
normal sections through the points where the axis meets the oblique sections. Ifn
is uneven, this will not be the case, but the deviation will be small for all the jets
investigated in this work. The only error to be considered will thus result from the
circumstance that the volume of the jet which is cut oft' by two normal sections, at a
constant distance from each other, will vary a little with the position in relation to
the stationary waves of the jet. To investigate the amount of this error the volume
of the jet V0L between the planes z = 0 and z = L is determined :

f$=2ir (V=I,
^
4=0 J*=ii

= 77-L

4/

If X is the wave-length, then k = 27T/X and

V0L = 7T . L .

~ X sin £ L

4 / 16 X

(9).

(10).

If X is equal to the distance between the edges (this distance is always greater, the
actual error consequently smaller than that calculated), then equation (10) shows
that the greatest volume that can be cut out is

whilst the average value is

= XTT (a2

INVESTIGATED BY THE METHOD OF JET VIBRATION. 361

The greatest error is, expressed in per cent.,

A =
For

b/n = 0-1 0-2 0-3 0-4

A = 0-04 0-16 0-35 0'65.

For the values of bja here used this error is without importance.
For the "jet-catcher" used here

L, = 2-97985 cm., L, = 2'99225 cm,, L = Li + L, = 5'9721 cm.

The whole of the "jet-catcher," with the exception of the before-mentioned steel
plates, is made of magnalium, which is both light and keeps in good condition. Before
use the "jet-catcher" is dipped in melted paraffin, so that it is covered with a thin
layer, removed only from the edges of the knives, Ic.

In order to prevent evaporation during weighing it is necessary to cover the
openings of the "jet-catcher"; this is done by the help of two indiarubber plates
fastened to two metal rods, pressed against the two openings by help of springs. It
is also necessary to introduce a correction for evaporation during the cutting off, as
described below.

The measuring itself takes place in the following manner : The fluid contained in
the "jet-catcher" from a previous measurement is completely removed and the apparatus
is carefully cleaned. The indiarubber coverings are set and the "jet-catcher" is
weighed and arranged on the pendulum, whereupon the rubber covers are removed
and the pendulum carried up to a horizontal position, released and allowed to
complete five whole swings, after which it is caught again. As soon as this takes
place the rubber covers are put on, the "jet-catcher" is taken from the pendulum and
all outside drops removed, after which the weighing takes place. After the conclusion
of the remaining measurements, which takes place in the course of a few minutes, the
jet is stopped and the " jet-catcher " again arranged on the pendulum. The rubber
covers are removed and the pendulum is allowed again to complete five whole swings,
beginning with the same height as with the " cutting off," after which the rubber
plates are replaced and the "jet-catcher" weighed again. If we call the weight of the
liquid contained in the "jet-catcher" by the first weighing P mg., and by the second
P— p mg., then the loss by evaporation during these swingings is p mg. The whole
weight of the quantity of the liquid cut off the jet is then P + a. p mg., as the loss by
evaporation during the cutting off is op mg.

The determination of the coefficient a takes place as follows : Two cuts are made,
the pendulum completing only one whole swing. The quantity of liquid " cut off" is
determined in the ordinary manner, after which the loss of weight, plt for five whole
swings of the pendulum is determined as explained above. Next four cuts are made,

VOL. ccvn. — A. 3 A

362

MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

the pendulum completing two whole swings, the corresponding loss, ps, being
determined in the same manner as above. In a similar manner p3, />«, and pb are
determined. We have then

In Table II. are arranged the results of these determinations for water and alcohol
(98*04 per cent).

TABLE II.

Water.

Alcohol,
98-04 per cent.

nig.

mg.

J»l

1-75

17-0

p*

2-30

22-0

P»

2-35

23-4

ft

2-40

23-6

Ps

2-45

24-0

3.

0-882

0-881

In the following it is always assumed that a = 0'88. A small error in the
determination of a is of no great importance. If the worst case in this paper
is taken, P is almost equal to 230 mg. and jps = 25 mg. ; we have then
P + op5 = 252 mg. An error in a of 0'04 will give an error in the concluding weight
of 1 mg. or of 0'4 per cent. The corresponding error in the surface-tension is about
0'2 per cent.

As the determination of p^ always takes place under the same conditions —
temperature, humidity, and air pressure — as those under which the " cutting off"
takes place, the determination of this correction is quite certain and cannot cause
great errors.

In Table III. are shown some of the values found for a.j)6 corresponding to orifice
No. III. All the weighings are corrected for the buoyancy of the air.

Besides the sources of error investigated there are several other circumstances that
possibly could cause irregularities in the exactness of the measurements. Thus it is
necessary that the vessels K of the "jet-catcher " have a certain shape, so that they can
without loss receive and hold the portions of the jet cut off. With the form shown
in fig. 6 I have never noticed any loss of liquid.

It is further obvious that if the speed of the "jet-catcher" when passing the jet is too
slow, the disturbance in the jet produced by the first knife will have time to reach
the second knife before it has cut the jet through. It is also possible that the
movement in the air resulting from the movement of the pendulum and the " jet-catcher "

INVKSTK;ATKI> BY THK MFTHOD OF JET VIBRATION.

363

TABLE TIL

Liquid.

*

Water

mg.
1-8

Alcohol, 3-09 per cent. l>v weight . . .
9-50 „ ...
46-34 . . .
74-93 „ ...
81-02 „ ...
90-97 „ ...
, 98-04 „ ...
Aniline

2-2
3-5
10-4
12-6
14-5
16-5
21-1
0-6

Ammonia, density pi 5/4 = 0-9903 . . .
„ „ = 0-9792 . . .
= 0-9580 . . .

2-6
5-0
7-9

might have influence upon it. It is clear that both these influences are dependent
upon the velocity of the "jet-catcher."

In order to investigate these questions I have made several series of experiments,
one of which is given in Table IV. The result is, as can be seen, within wide limits,
independent of the velocity of the "jet-catcher."

Further, I have compared the "jet-catcher" used here with another for which
L = Li + Lj = 7 '9030 cm. The difference between the results of a series of experi-
ments on the same jet was only 0*06 per cent.

TABLE IV.— Water Jet. Velocity 273-1 cm./sec. Diameter 1-3415 mm.

Mean velocity
of the
" jet-catcher."

Weight for five
complete oscillations
of the " jet-catcher."

Deviation from
mean value.

Deviation.

Corresponding
deviation of the
radius of the jet.

cm./sec.

mg.

mg.

per cent.

mm.

. 651

422-15

+ 0-83

+ 0-20

+ 0-00067

58G

421-04

-0-28

-0-07

-0-00024

530

421-43

+ 0-11

+ 0-03

+ 0-00010

463

420-80

-0-52

-0-12

-0-00040

382

421-97

+ 0-65

+ 0-15

+ 0-00050

280

420-52

-0-80

-0-19

-0-00064

Mean value . .

421-32

Mean error . .

±0-14

±0-00047

§ 7. After the above there can hardly be any doubt that this method for the
determination of the sectional area of a thin jet gives very trustworthy results, and
that by this means we have a convenient method of carrying out several investi-

.3 A 2

364 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

gations on such jets with an exactness that hitherto has been difficult or impossible

to reach.

I will mention only very briefly some measurements of the influence of the pressure
on the sectional area of a jet, keeping myself within the limits where the question is
of interest for this investigation. The measurements comprise two circular apertures,
No. 1 with diameter T514 mm. and No. 2 with diameter 0'8043 mm., both arranged
as shown in fig. 10, where B is the perforated plate. For these I have determined
the sectional area of water jets for heads between 50 and 100 cm. The results are
shown in fig. 8, where the value a-b corresponds to aperture No. 1 and c-d to

1,4
1,2
1,0
0,8
0,6
0,4
0,2

c-

0 10 20 30 40 50 60 70 80 90 100 cm. H.

Fig. 8.

No. 2. The values measured are shown by a cross ; it can be seen that they very
nearly fall on the straight lines a-b and c-d.

For aperture No. 1 the sectional area decreases 4'98 per cent., while the head
increases from 50-100 cm.

INVESTIGATED BY THE METHOD OF JET VIBRATION. 365

For aperture No. 2 the sectioual area decreases 5 '03 per cent., while the head
increases from 50-100 cm.

The apertures used in this investigation have diameters hetween No. 1 and No. 2.
For these, without making any great error, it can be calculated that the sectional
area decreases 1 per cent, for each 10 cm. the head increases.

The determination of the discharge takes place in the usual manner and needs no
comment.

Production of the desired Deviation from the Cylindrical Form of a Jet.

§ 8. In § 3 is shown the importance of the jet executing one single vibration, in
other words, that its surface is determined approximately by

r = a + l> cosn<£. cos (2;rz/X,) ........ (l),

as in the contrary case the determination of the wave-length A,, causes difficulty and
becomes inaccurate. By the measurements that up to now have been made with this
method only little attention has been paid to this condition. Lord RAYLEIGH* writes
as follows : "... The first set of observations here given refers to a somewhat
elongated orifice of rectangular form ; . . . refers to an aperture in the form of an
ellipse of moderate eccentricity ; . . . relate to an orifice in the form of an equilateral
triangle with slightly rounded corners. . . ." PiccARDt says : " Le liquide s'e"coule
par uu tube aplati. . . ." MEYER^ expresses himself in the following manner on this
question : " Die elliptische Oeffnung ist mittelst einer Stopfnadel durchgeschlagen,
welche auf einem Oelstein solange geschliffen wurde bis eine in ein Probestiick des
Membran geschlagene Oeffnung die gewiinschte Form und Grosse hatte. . . . Um
eine grossere Genauigkeit zu erzielen, ware vor allem auf eine schiirfere Beobachtungs-
metode und die Herstellung einer genau elliptischen Oeffimng Bedacht zu nehmen."

Apart from the last-mentioned reference — which according to § 3 is incorrect — the
importance of, and the means for, giving the jet a single vibration has been wholly
neglected.

I have endeavoured to solve this question by making the aperture as exact as
possible after the formula

<j> .... ...... (2).

As the dimensions must be small, for example a = 0*65 mm., so as not to use too
great a quantity of liquid, the work is consequently accompanied with some difficulty.
The following method, however, proved itself to be good. The aperture is first
drawn enlarged, fig. 9, ABCD, after which I chose a fine round file, the radius r

* RAYI.EIOH, ' Roy. Soc. Proc.,' 29, p. 71, 1879 (' Papere I.,' p. 377).

t I'lCXURD, IOC. tit.

| MEYEU, lot. cit.

366

MR. P. O. PEDERSEN ON THE SURFACE-TENSION OP UQUIDS

of which is somewhat smaller than the smallest radius of the curvature of the aperture.
On the drawing is constructed the curve abed, described by the centre of a circle with
radius r rolling inside the curve ABCD. The values of the radius vector for the
curve abed corresponding to <f) = 0°, 6°, 12°, &c., are determined on the drawing. By

Fig. 9.

help of these values and by using a small milling machine, with the above-mentioned
little file as cutter, the orifice can be cut in the correct form ABCD. Further
particulars are given in the original paper.

The plate B is in most cases of platinum iridium (90 Pt+10 Ir) and has the form

A

B

•KSK

pft<yj-Ti

B

*"yj.

-f.r'rVrrrfr " «-=*-* — -i ' f. — " . t • -," t , -r, i .--..•- — * a*

Fig. 10.

of a circular plate, about 17 mm. in diameter and about 0'5 mm. thick. In the
middle of each plate the thickneas is reduced to about 0'25 mm. Fig. 10 shows the
ordinary arrangement of aperture and conducting tube. A represents a glass tube,
B a perforated plate, and C a ring of iudiarubber. Some few apertures are made in

INVESTIGATED BY THE METHOD OF JET VIBRATION.

367

brass. Microphotographs of some of the orifices used are shown on Plate 2 ; below
each photograph is denoted the length of the largest diameter of the corresponding
orifice. Further particulars will be given later.

It proved, however, that even with the best of the apertures produced in this manner
the jet was not quite free from alien vibrations. That is due partly to deviations
from the correct form of the aperture, but also to the fact that the cross-section of the
jet is not strictly similar to the form of the aperture. This last-mentioned incon-
venience would be got rid of by allowing the jet to flow out of a tube which had the
correct form of cross-section. This solution is, however, for several re.osons incon-
venient. When the jet flows out of a tube the velocity will be less at the surface
than in the axis ; and, finally, the production of such a tube would l>e very difficult.

I have also tried to produce the deviation of the jet in another manner, namely, by
using a circular orifice and a non-circular conducting tube (see fig. 11) ; but generally
I prefer the other method.

B-

m <L

r . A

! S«^iJtea^^iEiV^}j?^H^;/«;

^V?-:v^.j.:f;;r,,%';-;;;';^>;^;-^;-;^.!^

> —

J_i

10cm.

Fig. 11.

With regard to the purity of the vibrations obtained, the jet-photographs on
Plates 3 and 4 will give good information. The production of these jet-photographs
will be described in the next section.

Determination of the Wave-length.

§ 9. Of all the quantities on which the surface-tension according to equation
[(2), § 1] depends, X, is undoubtedly the most difficult to determine. In all the
previous measurements, as mentioned Wore, \m is determined as the distance between
the summits of the jet, and the determination has taken place by direct measure-
ment either on the jet itself or on a photograph of it. As the amplitude of the
vibrations must be small, this method is very unsatisfactory and cannot give good
result*

An exact determination of X. can be made in many ways, but they will most
probably have it in common that the jet itself is used as an optical, image-forming
system. Of the methods I have endeavoured to use I will only describe the following

! \\'«.

368

MR. P. O. PEDEKSEN ON THE SURFACE-TENSION OF LIQUIDS

The first method is illustrated by fig. 12. Here abba represents one of the
profile lines of the jet (seen from above). L is a Nernst-lamp (1 amp. 220 volts),
the linear filament being vertical. The rays coming from L are reflected by the

Fig. 12.

mirror S and unite after reflection from the surface of the jet in the image Lt.
When the profile line is a sinusoid the distance between the images L' and L! will be
equal to the wave-length. This distance can be determined with great accuracy, and

ai

A|

•"""•"••n"1 ~

•ft

VLx

B bj C

i

^si

Fig. 13.

if the jet were perfectly free from alien vibrations this method would be able to give
very exact results. Unfortunately it has not been possible for me to produce a jet
so regular that I could make use of this method with real advantage. Even very
small deviations from the desired jet-form change the position of the images very

INVESTIGATED BY Till: METHOD OF JET VIBRATION.

309

considerably. The execution of a measurement also demands much time, and this is
probably the greatest drawback of the method.

The other method is as follows :—

The rays from a horizontal linear incandescent lamp L, (about 23 cm. long,
25 candle-power, 110 volts) is reflected from the mirror S perpendicularly down on
the jet * (see fig. 13). Close beside the jet is arranged a vertical photographic
plate P, upon which an image is formed, the approximate form of which is
shown by the line m-n on fig. 14. The lamp LI is enclosed in a shield ABC.

Fig. 14.
The part BC of this shield has the form shown in fig. 1 5, making the illumination

Section a-b.

Fig. 15.

of the jet the same for its entire length. In series with the lamp LI another lamp, I ...
is inserted, the power of which can be regulated by the help of a rheostat placed
parallel to it. I ... is arranged at the same height as the jet, and the light from it
produces a homogeneous fog on the plate P which is only interrupted by the
shadows st-sa of the jet and T,-Ta of the wire T which is arranged horizontally in
front of the plate.

The manner in which the image ni-n (fig. 14) is formed will be explained only very
briefly. In fig. 1 6 is shown the circular cross-section of a horizontal jet. L is a vertical
ray ; its direction is after two refractions and one reflection changed to 1^. The angle
between L and L, is denoted by y and the refractive index of the liquid by n^ With
the syinlxils of the figure

y = 4/>-2t.

VOL. CCVII. — A. 3 B

370 MR, P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

It is easily shown that y is maximum when

sn =

In order to illustrate the positions of the emergent rays, fig. 17 is drawn. The
ray for which y is maximum cut the plate P in B, and at that point the intensity of
illumination will be maximum. All the points B, collectively, form the image m-n
(fig. 14).*

Let the maximum value of y corresponding to zf be denoted by y,, then y,-y0 has the
same sign as bn cos kz. The wave form of the image m-n is produced in this manner.

P

Fig. 17.

As the amplitude of the curve m-n is much greater than that of the jet, it is
much easier to determine the wave-length by measuring on the image m-n than on
the jet itself.

The measuring of the wave-length takes place in the following manner (fig. 14) :

"The"" -distance L between two homologous suitable points on the image m-n is

determined. By dividing this length by the number v of waves between the points

* For further information "about this question see J. M. PERNTIIER ' Meteorologische Optik' (Wien and
Leipzig, 1902), p. 482.

t In r

cos n<t>. cos kz . . . [(1), §1].

INVKSTICATKI) BY THI. MKTHOD OF .TET VIBRATION. 371

measured, we have X,. This would be perfectly correct if the jet were horizontal over
the whole length, in other words, parallel to T,-T,, and perpendicular to the light
incident on the jet. As this is not the case, the following correction is necessary :
L represents the distance hetween the homologous points in the image, hut what is in
reality necessary to l>e known is the distance between the corresponding points p and y
on the jet itself. With the symbols used in the figure, this correction for the point

p will be with sufficient exactness : x = -,[ab + ac— (ai

Here e is the distance from the point on the image to the point Q (see fig. 17)
where the ray of minimum deflection is reflected. This correction is calculated for both
the two points p and </, and the distance L, Ixjtween these points is L, = L— x— x,.

This formula is not quite correct, as e as a rule will have different values at the
two ends, but the corresponding error is only small, and will be neglected here. L,
is here determined as the distance between the points p and </, although in reality it
is the length of the portion />— 7 of the jet that is needed ; but this error is only
small for the jets examined here.

The wave-length is therefore determined by X, = L,/^.

In the following, the wave-length is always determined by the last-mentioned
method, although perhaps it is not so exact as the first, in principle ; it has never-
theless great advantages compared with it. Among these advantages is the
comprehensive view of the whole jet, tending to prevent mistakes, and the much
shorter time needed for the determination, inasmuch as the actual measuring work
can be done afterwards on the finished plate. Finally, the exactness that is reached is
certainly as great as is possible, so long as it is not feasible to obtain absolutely pure
jet-vibrations. One fault, however, with this method is that it is only available for
transparent liquids.

1 hiring exposure the plate P is arranged in a plate-holder which is fixed in a
vertical frame. This can be laid down in a horizontal position by turning the pivots
below. The frame is arranged on a horizontal slide that can move in a direction at
right angles to the jet. The movement of the slide towards the jet is stopped by an
adjustable stop leaving a distance of about 4 mm. between the plate and the jet.

< Mi Plates 3 and 4 are shown some photographs of jets taken in this manner ; further
details will l>e given later.

By the use of nearly monochromatic illumination still tatter jet-images may be
obtained.

Inr< xtiti'ition* on the Influence of tin- Amplitude of Vibration,

;; 1 0. If the jet's cross-section is determined by the equation

r = a + b cos n«f> ....... . ., % . (1),

then rnu,x = a + b and /•„,,„ = a— b.

3 B 2

372 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

In the following the amplitude of the jet will be denoted by 8, determined by

or, according to equation (1),

8 = 100. m" m"1 ... (2),

'max ~r mln

8=100.* . . . -. . . (2').

a

The same notation will be used for the orifices.

In § 1 the necessity of an investigation respecting the influence of the amplitude
on the period of vibration has been already emphasized. The only material that is
available in this case consists of the measurements made by Lord RAYLEIGH* and
recorded in his original paper. For some of these I have calculated the surface-tension
according to formula [(2), § 1], and arranged Table V. in order of decreasing amplitude.

TABLE V.

Orifice.

T.

Triangle 1

ivith slightly rounded corners. (Table V., by RAYLEIGH) . . .

dyne/ cm.
57-4

Square

(Table VII , by RAYLEIGH) . . .

64-6

Ellipse of

moderate eccentricity (S > 9). (Table II., by RAYLEIGH) . .

69-5

Ellipse (5

- 9). (Table IV , by RAYLEIGH)

72-9

From the values of T it is evident that the amplitude for the three jets has been
too great. How far this is also the case with the jet corresponding to the orifice for
which 8 = 9 cannot be determined on the basis of the investigations mentioned. In
the measurements of MEYER! and PICCAKD,^ the amplitudes in accordance with the
above have been much too great.

In order to decide this question, I have made a series of measurements with jets of
water ; the orifices used for these are recorded in Table VI.

The results of these experiments are found in Table VII., where the orifices are
arranged with decreasing amplitudes. In the table, T indicates the surface-tension
calculated upon the supposition that the amplitudes could be considered as extremely
small.

* RAYLKIGH, 'Roy. Soc. Proc.,' 29, p. 71, 1879 ('Papers I,' p. 377).
t MEYER, loc. cit.
} PICCARD, l«f. at.

INVESTIGATED BY THE METHOD OF JET VIBRATION.

373

TABLE VI.

Diameter of orifice.

Orifice No.

V.

5.

Material of perforated
plate.

Largest.

Sni.-illvflt.

I.

mm.
1-671

mm.
1-009

2

24-7

90Pt+10Ir

II.

1-545

1-281

2

9-3

90 Ft +10 Ir

III.

1-424

1-303

2

4-4

90Pt + 10Ir

IV.

0-830

0-820

•>

0-61

90 l't + 10 Ir

V.

1-386

1-384

•-'

0-14

90 Pt + 10 Ir

VII.

1-556

1-530

1

0-86

90Pt+10lr

A

2-202

_

3

brass

B

2-372

2-128

4

—

99

C

2-398

2-318

6

~

It

TABLE VII.— Ordinary Tap- Water.
Pw = 0-99913, = -0-151.

Experi-
ment

No.

Orifice.

Dis-
charge
of the
jet.

Sectional
area of
the jet.

Wave-
length.

T,.

/.

M--

T»

No.

n.

8.

om'./wo.

cm1.

cm.

djne cm.

dyne rm.

dyne cm.

18

I.

2

24-7

2-9508

0-01060

1-2100

65-63

16-4

o-oo

65-85

17

IL

2

9-3

3-1864

0-01149 1-2306

70-99 17-0

o-oo

71-29

28

II.

2

9-3

4-6535

0-01097

1-8210

72-38 14-3

o-oo

72-27

19

III.

2

4-4

2-9942

0-01073

1-1506

74-02 14-6

o-oo

73-96

29

III.

2

4-4

4-3291 0-01020

1-7041

74-07 14-4 0-00

73-97

37

Ill

2

4-4

3-8649 0-01043

1-5095

73-99 14-5 0-06 73-8U

38

HI.

2

4-4

3-8327 0-01043

1-4932

74-33 15-1 0-07

74-27

39

III.

2

4-4

3-8597 0-01049

1-5053

74-18 15-4

0-06

74-18

23

VII.

2

0-86

3-8745 0-01394

1-3945

74-40 15-2

o-oo

74-43

33

VII.

2

0-86

5-6274 0-01334 2-0675

74-53

14-5

o-oo

74-45

20

IV.

2

0-61

1-2580 0-004728

0-5800

75-03

14-8

o-oo

75-00

30

IV.

2

0-61

1-8440

0-004551

0-8815

74-29

14-3

o-oo

74-18

21

V.

2

0-14

3-0412

0-01097

1-1612

74-12

14-9

o-oo

74-11

31

V.

2

0-14

4-4577

0-01060

1-7387

74-03

14-0

o-oo

73-88

24

A

3

8-2429

0-02912

1-2620

71-03

15-0

o-oo

71-03

25

B

4

._„

8-8476

0-03113

0-8222

73-40

14-9 0-00

73-39

26

C

6

9-3313

0-03312

0-4467

74-83

14-2 0-00

74-70

The values of T for the orifices A, B, and C agree very well with those for the
apertures for which n = 2, when due regard is taken to the amount of the amplitude.
In the following, notice is only taken of those orifices for which n — 2.

374 MR. P. 0. PEDERSEN ON THK SURFACE-TENSION OK LIQUIDS

TABLE VIII.

Orifice.

8.

Mean value
of T,».

74-34-T15.

74-34-T,6

0-02. «2.

8*

I.

dyne/cm.

24-7 65-85

djne/cm.
+ 8-49

0-0139

12-2

II.

9-3 71-78

+ 2-56 0-0292 1-73

III.

4-4

74-05

+ 0-29

0-0155

0-39

VII.

0-86 74-44 -0-10

0-015

IV.

0-61 74-59 -0-25

0-008

V.

0-14

74-00 +0-34

|

—

o-ooo

VII., IV., V.

—

74-34

—

—

In Table VIII. are given the mean values of Tu corresponding to each aperture. It
is evident that the orifices I. and II. have too great amplitudes. For the orifice III.
it cannot with certainty be determined on the basis of Table VIII. ; if the amplitude
has any influence on the period it can only be said that the influence must be small.
The amplitude for the next orifice, VII., is only one-fifth of that of orifice III., and
there can therefore be no doubt that the amplitude of orifice VII. is sufficiently
minute. That is still more certain for the orifices IV. and V. In Table VIII. is also
inserted the mean value of T16 for holes VII., IV., and V. taken together. This mean
value is 74'34. The fourth column contains the values of 74'34— T16 and the fifth

7 J. -^ A. T1

column contains the values of ' — ^ — - for the orifices I., II., and III. Finally in

the last column the values of 0'02 . S2 are given for all the orifices. The numbers in
the last two columns must naturally be taken with all possible reserve, but still they
serve to explain, and at the same time also prove, the correctness of the above result,
namely, that the amplitudes for holes VII., IV., and V. are so small that their values
have no appreciable influence on the determination of the surface-tension.

Reference has only been made in the above to the amplitude of the aperture and

TABLE IX.

Orifice.

Pressure.

8 measured on
the jet.

cm*
I. 95

37-6

I.

43

29-2

II. 95

16-3

III.

43

4-9

VII.

43

1-06

IV. 43

0-88

INVESTIGATED I!Y TMK MKTHOD OF JET VIBRATION.

375

not to the jet itself. To get an idea of the size of the amplitude of the jet I have
taken photographs of different jets, and afterwards by help of an object-micrometer
measured the largest and smallest diameters at a distance of 3 to 4 cm. from the
apertures. The results of these determinations are given in Table IX. ; it appears
that the amplitude of the jet is somewhat greater than the amplitude of the orifice,
and that it increases with the pressure.

Therefore the above conclusion respecting the permissible amplitude will not always
hold good. The pressure, about 70 cm., used by the measurements made here, lies,
however, within the limits investigated in Table VII. (alwut 42 up to 97 cm.
pressure).

The nature of the liquid may also play a part ; but that has hardly any great
influence as far. as these investigations are concerned.

In order to attain greater certainty on this point I have determined the surface-
tension of some other liquids by measurements with orifices III., V., and VII. ; the
results obtained are shown in Table X. This is calculated as follows : In the same
manner as in Table VII., TI8 is determined for each individual orifice, and for each of
these the mean value is taken. The mean value for all the measurements with
apertures VII. and V. is taken as the correct value of the surface-tension for the
liquid in question. Table X. contains the deviations from the mean value found in
this manner, with reference to the orifices in question, shown as a percentage of the
surface-tension.

TABLE X.

Liquid.

Orifice III.

Orifice V.

Orifice VII.

Ordinary tap-water
Distilled water

per c«nt.
-0-23
-0-54

per cent.
-0-31
-0-44

per cent.
+ 0-30
+ 0-44

CuSO4 + Aq; t> - 1-0506

-0'34

-0-17

+ 0-19

5 -79 per cent, alcohol + 94 -21 per cent, water

-0-90

-0-07

+0-00

Mean value

-0-60

-0-25

+ 0-23

It appears that the results for orifice VII. are generally a little larger than for
orifice V. The difference lies, however, within the limit of error, as the deter-
mination for orifice V. is difficult. It proves, however, that under the present
conditions the amplitude for orifice VII. is sufficiently small. On the other hand,
however, it also appears that the values for orifice III. are, as a rule, a little too
small. It would therefore be natural to carry out the measurements with orifices
VII. and V. The determination is, however, in reality made with orifices VII. and
III., for the following reason: The amplitude for orifice V. is so small that the
determination of the wave-length is difficult and uncertain, especially for liquids with
small surface-tension, or, in other words, large wave-lengths. On the other hand,

376 MR. P. O. PEPERSEN ON THE SURFACE-TENSION OF LIQUIDS

the measurements of the wave-length for orifice III. are carried out with great exact-
ness and easiness, just as the determination with orifice VII. is, as a rule, quite good.
I have therefore chosen these two orifices and corrected for the too great amplitude of
orifice III. by adding 0'5 per cent, to the results, a correction obtained from Table X.
In the following this correction is represented by [8].

Execution of Observations.

§ 11. All the experiments to No. 37 are made with ordinary tap-water under
constant pressure, produced in the manner explained in § 4. In all the other experi-
ments the pressure diminished as the liquid ran out. This variation was, however,
only small, as the cross-section of the reservoir used was about 400 cma. and the
quantity of liquid used about 1000 cm3. The pressure for all the experiments after
No. 40 was about 70 cm.

The measurements themselves took place in the following manner :—

The orifice is closed by a wooden plug. The requisite quantity of liquid is poured
into the reservoir, care being taken to fill both the conducting tube and the jet tube
completely, the plug is withdrawn and the jet started. Then the jet is adjusted so
as to be parallel to the plate-holder and to have the height suitable for the
"jet-catcher." The last adjustment is easily controlled by the shadow of the jet on
a ground glass placed in the frame and furnished with marks, between which the
shadow must fall. At the same time the jet tube is moved until the image (m-n,
fig. 14) of the jet is as sharp and clear as possible. In order to enable these
adjustments to be made easily, the jet tube is arranged in a bridge that can be moved ^
in all directions by help of screws.

As the direction of the "jet-catcher " once and for all is parallel to the frame, the jet
by the above adjustments is brought into the required position and the measuring of
its cross-section can take place. Immediately afterwards the measurements of the
discharge begin, after which the plate-holder with an unexposed plate is placed in
the frame, which is lying down. The light is then shut off, the shutter removed
from the plate- holder and the slide moved into its position, whereupon the frame is
brought up to its vertical position and the lamps Lt and L2 lighted. After exposing
for about 15 seconds the lamps are turned out, the slide brought back and the
shutter replaced in the plate-holder. The entire photographing process takes about
40 seconds. Before finishing the measurement of the discharge another photograph
is taken in the same manner.

Finally the necessary weighing takes place and the evaporation is determined as
explained in § 6.

In changing from one liquid to another the whole apparatus is cleaned very
carefully and finally washed out with distilled water, after which it stands for some
time to dry. Before use it is washed out with some of the liquid to be tested.

INVESTIGATED BY THE METHOD OF JET VIBRATION. 377

Various Remarks.

§ 12. On account of the pressure not being constant, it is necessary to investigate
the influence of its variations.

According to § 7 the cross-section of the jet increases about 1 per cent, for every
10 cm. the pressure decreases. The cross-section ought, then, to be measured at mean
pressure, but is in fact determined at the commencement of the experiment. If the
liquid pressure has diminished h cm. during the experiment, the mean cross-section is

;

A +

2000 '

where A is the measured cross-section. The corresponding correction [e] in the
surface-tension T is, then, with sufficient exactness

This correction is always negative.

Influence of the Variation of Pressure on the Wave-Length and Discharge. — If the
effective pressure reduces from H cm. to H— h cm., the first photograph of the jet
will correspond to the pressure H— y cm. and the last with sufficient exactness to the
pressure H— h + y cm. The corresponding velocities are

Va =

The mean value is

, : VaA = i[v/2^(H^T) + v/27(H^F^J] ...... (2).

As y and h are small compared with H, this expression can without any appreciable
error be reduced to

Va4=^[v/2^H + v/2flr(H-Aj] ....... (2').

As the wave-length is determined as the mean of the results from the two plates, V^ is
the velocity corresponding to the wave-length measured. The average velocity V0
that determines the discharge Q is, as is known, similarly determined by

V0 =

In this manner no correction is demanded on account of variable pressure in the
ili •termination of the wave-length and the discharge.

The curvature of the jet produces a small error since the cross-section is determined
for the highest part of the jet. The average cross-sectional area will therefore be
a little smaller than that measured. Under the conditions used here this error will
only be insignificant.

VOL, CCVII. — A. 3 C

378 MR. P. O. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

The influence of the viscosity is, according to equation [(4), § 1], determined by the
coefficient

' - . '- • - (3),

where 8 is the logarithmic decrement oi the vibration. The determination of 8 is
made by measuring on the image (m-n, fig. 14) of the jet, making the supposition
that the amplitudes of the image are proportionate to those of -the jet. The results
are—

For water : 8 = 0'074 ; g = 1 '00014 ;

„ 98-04 per cent, alcohol + 1'96 per cent, water : 8 = 0'173 ; g = 1 "00076 ;
„ 46'34 per cent, alcohol + 53'66 per cent, water : 8 = 0'210 ; g = I'OOllO ;
„ aniline: 8 = 0'265 ; g - 1'OOIS.

The numbers given are only approximate, but they show that g, in all cases
considered here, is almost equal to 1, and as the determination of 8 is uncertain, no
correction is introduced. An exact investigation of the influence of the viscosity on
the form of the jet image cannot be made until the theory of this image is further
developed.

Remarks on the Jet Photographs.

§ 13. On Plates 3 and 4 are shown fifteen jet photographs. In each photograph
the number of the orifice and the nature of the liquid are denoted.

The photographs are arranged according to the orifices, in the same order as in
Table VII., and the remarks concerning them are given in the same order.

Plate 3, figs. 6, 7 and 8, shows clearly the influence of viscosity on the damping of the
vibration. Fig. 6 is a water jet : this has only a very small damping. Fig. 8 is an
alcohol jet : with this the damping is a little larger. Fig. 7 is a jet of a mixture of
water and alcohol (46'34 per cent, alcohol + 53'66 per cent, water): with this the
damping is much greater than for the other two. The viscosity has about the follow-
ing values in these three cases (Tn. GRAY, ' Physical Tables,' Table 151, 1897) :—

, 0'012 by 15° C. : , 0'036 by 15°C. ; , 0'014 by 15°C.

As the logarithmic decrement is the same for all vibrations, the fundamental
vibration will be purer at some distance from the orifice than immediately after
the jet has been formed, as is also shown on several of the photographs.

It appears from these photographs that the jet image is very well adapted to
investigation of the jet vibrations. Similarly, they show that it is possible to make
orifices which for all practical purposes are correct. In further investigations by this
method it will be possible to go still farther in this direction.

All the jet photographs commence about 1 '5 cm. from the orifice.

INVESTIGATED BV THE METHOD OF .IET VIBRATION.

379

RESULTS.

Water.

§ 1 4. For the determination of the surface-tension of water I have made three series
of measurements, the results of which are shown in the Tables XL, XII., and XIII.
The first table refers to ordinary tap- water, and gives a mean value

T,» = 74-33 dyne/cm.

The secoml tahle is for freshly distilled water, and gives

T,» = 74-31 dyne/cm.

The third table is for distilled water that has been kept for about a year in a
stoppered Iwttle : the result is

T,» = 74-23 dyne/cm.

The greatest value found is in experiment No. 44 (Table XIII.) :—

Tlft = 74-80 dyne/cm.
The least is in experiment No. 41 (Table XIII.), namely :—

Tlft = 73-40 dyne/cm.

The greatest deviation in the 18 experiments recorded in the Tables XI. -XIII. is
thus about 1'9 per cent.

TABLE XI.— Ordinary Tap- Water.

irp

plb!4 = 0-99913, '- = -O'lol.

Experi-
ment
No.

Orifice.

Discharge
the jet.

Sectional
area
of the jet.

Wave-
length.

T,.

/.

[']•

[«]•

Tu.

19

III.

cm'.'seo.
2-9942

cm*.
0-01073

cm.
•1506

dvne 'cm.
74 -02

°0.

14-6

dyne/cm, dyne/cm.
0-00 0-37

dyne/cm.
74-33

29

III.

4-3291

0-01020

•7041

74-07

14-4 0-00 0-37

74-35

37

III.

3-8649

0-01043

•5095

73-99

14-5

0-06 0-37

74-22

38

III.

3-8327

0-01043

•4932

74-33

15-1

0-07

0-37

74-65

39

III.

3-8597

0-01049

•5053

74-18

15-4

0-07

0-37

74-54

23

VII.

3-8745

0-01394

1-3945

74-40

15-2

o-oo

—

74-43

33

VII.

5-6274

0-01334

2-0675

74-53

14-5

o-oo

—

74-47

21

V.

3-0412

0-01097

1-1612

74-12

14-9

o-oo

74-12

31

V.

4-4577

0-01060

1-7387

74-03

14-0

o-oo

—

73-88

Mean value of orifice III

74-42

74-45

74-00

„ „ all expcrimei

its

74-33

3 c 2

380 ME. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

TABLE XII.— Distilled Water.
Tested about two days after the distillation.

put = 0-99913, = -0-151.

Experi-
ment
No.

Orifice.

Discharge
of
the jet.

Sectional
area
of the jet.

Wave- T

length.

/.

[•]•

[4

TM.

129
130
131
132

III.
III.
VII.
VII.

cm'/'ec.
3-8514
3-8516
5-0647
5-0696

cm*.

0-01042
0-01043
0-01381
0-01380

cm. dvne/cm.

1-5038 . 74-07
1-5000 ! 74-41
1-8424 ; 74-36
1-8408 74-64

°C.
13-3
13-6
14-0
14-2

dyne/cm.
0-06
0-06
0-06
0-06

dyne/cm.

0-37
0-37

dyne/cm.
74-12
74-51
74-15
74-45

Mean value of ori

n i» »i

,, ,, all

fice III.

74-32
74-30

VII

experimei

its

74-31

TABLE XIII.— Distilled Water.
Tested about one year after the distillation. In the meantime kept in a corked vessel.

= 0-99913,

= -0-151.

Experi-
ment
No.

Orifice.

Discharge
of
the jet.

Sectional
area
of the jet.

Wave-
length.

T,.

t.

M-

c,

T15.

cma./sec.

cm*.

cm.

dyne/cm.

°C.

dyne'cm.

dyne/cm.

dyne/cm.

40

III.

3-7997

0-01037

1-4869

73-83

19 -b

0-06

0-37

74-74

41

III.

3-7450

0-01032

1-4808

72-50

18-9

0-06

0-37

73-40

42

V.

3-9079

0-01079

1-5224 i 73-05

18-6

0-06

—

73-53

43

V.

3-8899

0-01074

1-5068 ! 74-03

18-2

0-06

—

74-45

44

VII.

4-9899

0-01354

1-8226

74-39

18-1

0-06

74-80

45

VII.

4-9532

0-01358

1-8115

74-10

18-0

0-06

—

74-49

Mean value of orifice III

74-07

„ „ V

73-99

„ VII

74-64

„ „ all experimei

its

74-23

That the results are almost the same for the three kinds of water is not surprising,
as DUPR£* and Lord IlAYLEiGHf have shown that even very considerable impurities

* DUPR&, ' ThtSorie mecanique de la chaleur,' p. 376, Paris, 1869.

t RAYI.EIGH, ' Roy. Soc. Proc.,' 47, p. 281, 1890 (' Papers III.,' p. 340).

INVESTIGATED BY THK MKTHOD OF .TET VIBRATION. 381

do not appreciably alter the surface-tension, as far as quite fresh surfaces are
concerned.

As the result of my experiments I fix the initial value of the surface-tension of

water as

T15 = 74-30 dyne/cm.

The surface-tension of water has been so often determined, and in so many manners,
that even a moderately exhaustive representation of the results is impossible and,
besides, without great interest, as many of the measurements have but little value. In
Table XIV. are shown the results of a few determinations by the capillary-tube method,
and Table XV. contains most of the results obtained by the method of capillary ripples.

With respect to the values found by the capillary-tube method, it appears that,
with the exception of QUINCKE'S values, they are all smaller than those found here.
This is quite natural, because it is the stationary value of the surface-tension that is
measured by the capillary-tube method. Under the given conditions this value must
be smaller than the initial value.

The values found by the method of capillary ripples in most cases agree well with
the value found here. They are as follows : — Lord RAYLEIOH, 74*35 dyne/cm. ;
DORSEY, 7372 ; WATSON, 7476 ; and KALAHNE, 74'22 dyne/cm. The mean value

for all four is

T1S = 74-26 dyne/cm.

An exception from this, however, is made by GRUNMACH'S measurements (BRUMMEK
and LOEWENFELD, who worked exactly in the same manner as GRUNMACH, are not
mentioned here).

GRUNMACH'S measurements divide themselves into two groups, the surface of the
liquid being either the same during the investigation or continually renewed.
Table XV. shows that GRUNMACH'S value for distilled water in the first case is

T1& = 78-41 dyne/cm.,

and in the other

T16 = 75-89 dyne/cm.

These results are very extraordinary, for two reasons. Firstly, it would be expected
that the former value would agree with the values found with the same method by
other investigators. This, however, is far from being the case, as GRUNMACH'S value,
78*41 dyne/cm., is 5*6 per cent, larger than the corresponding mean value, 74'26
dyne/cm., of the other measurements.

Secondly, it would be supposed that the surface-tension for the continually renewed
surface would be the greater. GRUNMACH, however, came to the opposite result and
found a value 3 '3 per cent, lower in this case.

A satisfactory explanation of this circumstance will certainly demand fresh investi-
gations, and before these are finished it will be difficult to judge of the value of

382

MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

§*

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INVESTIGATED »Y THE METHOD OF JET VIBRATION.

388

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Authority.

RAYLEIOH

DORSEY

1

A. KALAHNE

A. KALAHNE

A. KAIJiHNE

A. KAIJiHNE

L. GRUNUACH

L. GRUNMACH

A IlKIMMIK

KOIX)WRAT-
THCHERWINSKI

K. LOKWKNKKI.U

384 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

GRUNMACH'S results. I cannot, however, omit to draw attention to the fact that the
mutual agreement between GRUNMACH'S values is only small. Within the first group
of measurements the deviation amounts up to 16 per cent., and for the last to 3 '9
per cent.

The lowest of GRUNMAOH'S values in the two cases are

T15 = 70'4 dyne/cm, and T^ = 74 '2 dyne/cm.
Both are lower than the values found here.

Toluol.
The value found according to Table XVI. is

TIJ = 2876 dyne/cm.,

in complete agreement with VOLKMANN'S results, namely,

TIS = 2879 dyne/cm.

This was to be expected, as in this case the initial and the stationary value of the
surface-tension must be very nearly equal.

Aniline.

The liquid used was marked " pure," but was, however, a little coloured.
The result of the measurements is shown in Table XVI. The value is

T15 = 43-00 dyne/cm.
The corresponding value by VOLKMANN is

T,, = 44-30 dyne/cm.,

also considerably larger. To this may be remarked that VOLKMANN himself estimates
his determination as somewhat uncertain, and, further, that aniline is somewhat
soluble in water, so that there has possibly been formed a layer of water on its
surface. Using the method of the maximum pressure of small air bubbles, FEUSTEL*

found - : TI§ = 46-6 dyne/cm. V ^ \ f/ '

Aqueous Solutions of Ammonia.

Measurements were -made on three solutions of ammonia. The values are (see
Table XVI.),

For /3151 = 0-99030, T16 = 71 '25 dyne/cm.,

» /°is,4 = 0-97921, T,8 = 68-02 dyne/cm.,
= 0-95801, T16 = 64-69 dyne/cm.
* FEUSTEL, loc. tit.

INVESTIGATED BY THE METHOD OF JET VIBRATION.
TABLE XVI.

385

Liquid.

Density.

Pun ~

Number
of
measure-

Greatest
deviation
from
the mean

Mean value.
T,&-

ment*.

value.

per cent.

dvne/oin.

Toluol

0-86736

2

0-07

28-76

Aniline

1-0250

5

0-50

43-00

•
AqiiooiiH solution of ammonia .

0-99030

4

0-66

71-25

0-97921

4

0-75

68-02

0-95801

4

0-70

64-69

Solution of copper sulphate . .

1-05030

6

0-53

74-27

Diluted sulphuric acid

1-08130

4

1-30

74-89

1-14316

4

0-89

74-44

Aqueous ethyl alcohol —

1 • 1 3 per cent, alcohol + 98-87 per cent, water . .

0-99702

4

1-05

69-65

3-09 +96-91 . .

0-99350

4

0-04

62-99

6-79 +94-21 . .

0-98910

5

0-55

56-66

9-50 +90-50 . .

0-98374

4

0-40

50-23

25-63 +74-37 . .

0-96328

4

0-89

34-98

37-88 +62-12 . .

0-94304

4

0-95

30-52

46-11 +53-89 . .

0-92631

4

1-50

28-07

46-34 +53-66 . .

0-92578

4

1-12

28-73

49-22 +50-78 . .

0-91952

4

0-98

26-57

50-99 +49-01 . .

0-91567

4

0-33

27-45

59-37 +40-63 . .

0-89841

4

1-28

26-55

74-93 +25-07 . .

0-86012

4

0-59

25-61

81-02 +18-98 . .

0-83522

4

0-90

24-57

90-97 + 9-03 . .

0-81973

4

0-46

23-82

98-04 + 1-96

0-79960

8

1-71

22-80

For comparison I will take the values found by DOMKE* by the capillary-tul>e
method. The values of DOMKE and those of the author agree fairly well, as the last,
as could be expected, are all a little larger than the first. If we compare DOMKE'S
values augmented with the difference between the author's and DOMKE'S results for
distilled water (namely, 74'30 — 73'00 = 1'30 dyne/cm.), it appears that the difference
between the two sets of values is not great (see Table XVII.).

LoEWENFELDf has determined the surface-tension of ammonia by the method of
capillary ripples with surface renewal. His results differ rather much from the
author's.

* DOHKE, 'Wiss. Abh. d. K. Norm.-Aich.-Komm.,' Heft III., Berlin, 1902.
t LOEWEXFKI.D, 'Diss.,' Berlin, 1905.
VOL. CCVII. A. 3 D

386 MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

TABLE XVII.

a.

1.

P.

PlS/4-

DOMKE.

DOMKE+1'30.

The author.

-c.

1-000

dvne/cm.
'73-0

dyne/cm.
74-3

dyne/cin.
74-30

dyne/cm.

o-oo

0-9903

69-2

70-5

71-25

-0-75

0-9792

66-5

67-8

68-02

-0-22

0-9580

63-3

64-6

64-69

-0-09

Solution of Copper Sulphate.
pm= 1-0503.

The mean value of the determination of the surface-tension of this solution is found

in Table XVI,

T,6 = 74-27 dyne/cm.

The value is practically the same as for water.

Diluted Sulphuric Acid.

Of diluted sulphuric acid two different concentrations were investigated. The
mean values are (see Table XVI.)

For plb = T0813, T15 = 74'89 dyne/cm.,
„ ptb = 1-14316, T15 = 74-44 dyne/cm.

The corresponding values, according to GRUNMACH (with capillary-wave method
with renewed surface), are

T1B = 76-5 dyne/cm, and TJ5 = 7 7 '8 dyne/cm.

Aqueous Ethyl Alcohol.

DOMKE* has given a table of the results obtained by different authors for the
surface-tension of absolute alcohol. The results are reduced to a temperature of
15° by the use of dT/dt = — 0'08. The mean value of the fifteen values considered is

T15 = 23-1 dyne/cm.

The lowest value is T,6 = 2 2 '2 dyne/cm., the highest is T15 = 24'3 dyne/cm.
DOMKE himself found by the capillary-tube method,

T15 = 23-0 dyne/cm.
* DOMKE, -Wiss. Abb. d. K. Norm.-Aich.-Komm.,' Heft III., Berlin, 1902.

INVESTIGATED HY Till MKTHOI) OF JET VIBRATION. 387

GRUNMACH* found by the method of capillary ripples :—

For absolute alcohol that has not been in contact

with the air and for a continually renewed

surface. Tl& = 19 '6 dyne/cm.

For absolute alcohol the surface of which had

been in contact with the air for one half-hour . T,6 = 2 1 '2 dyne/cm.
For absolute alcohol that had for some time been

continually in contact with the air .... T15 = 26 '3 dyne/cm.

The author determined the surface-tension of several mixtures of alcohol and water.
The mixtures investigated and the results obtained are shown in Table XVI. The
value of dT/e/£ used by the calculation of the table is determined by the following
formula : —

^ = -(0-151 -p. 0-0007),

where p is the percentage of alcohol by weight.

The results are shown in Plate 2, fig. 1, where also the results found by
B. WEiNSTEiNt are shown.

With regard to the surface-tension of absolute alcohol it can pretty certainly be
taken that the value determined by this method will very approximately be

T,6 = 22'5 dyne/cm.,

in complete disagreement with the value found by GRUNMACH, T18 = 19'6 dyne/cm.,
in the case of continuous surface renewal.

How this great difference is produced must be determined by further investigations
on the subject.

CONCLUDING REMARKS.

§ 1 5. In the use of Lord RAYLEIUH'S method for the determination of the surface-
tension of liquids it is necessary to pay attention to the following remarks : —

It is necessary to determine either the velocity or the cross-section of the jet by
direct measurement, as the calculation of the velocity from TORRICELLI'S formula may
lead to great errors.

The greatest care must be taken to obtain jets executing one vibration only,
corresponding to only one value of u.

The amplitude of vibrations must be very small.

The determination of the wave-length must be performed by suitable optical

* GRUNMACH, lot. at.

t WEINSTRIN, ' Metronomische Beitrage,' No. 6, Berlin, 1889.

3 D 2

888

ME. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

methods, as the smallness of the amplitudes renders the direct measurement
impossible.

With due consideration of the above remarks, Lord RAYLEIGH'S method is a very
good one, and is highly deserving of use in the future on account of its great
fundamental advantages.

With regard to the results obtained in this investigation, the author desires to call
attention to the remarkable discrepancies between his results and those of GRUNMACH,
who used the method of capillary ripples^with renewal of the surface (see p. 343
above). It was to be expected that the difference between the two sets of values
should be small, and further, that GRUNMACH'S values should be intermediate between
the author's values and those obtained by the capillary-tube method. But that is
very far from being the case : the differences are rather great, and of a sign opposite
to that expected.

In further application of this method the author would propose to use apertures
with amplitudes between 4'0 and 0'5 ; for instance, the following set : —

8 = 4-0, 8 = 2-0, 8 = 1-0, and 8 = 0'5.

TABLE XVIII.

Liquid.

Experiment No.

T15.

H.

/*15-

A.

dyne/cm.

cm.

cin.J

Toluol

156

28-75

67-4

0-0064

0-00994

Water

129

74-12

69-6

0-0114

0-01042

9 • 5 per cent, alcohol .

115

50-05

65-1

0-0175

0-01058

46 • 34 per cent, alcohol

89

28-93

59-2

0-0352

0-01131

81 '02 per cent, alcohol

71

24-63

63-8

0-0227

0-01075

98 • 04 per cent, alcohol

59

22-78

63-5

0-0135

0-01054

Aniline

52

42-86

57-8

0-0550

0-01156

Table XVIII. contains a few results compiled from the above measurements,
illustrating the relation between the cross-section of the jet (A) and the coefficient
of viscosity (/u,). The table is calculated for orifice III. ; H is the effective head.
The table shows that the cross-section increases with the viscosity.

INVESTIGATED BY THE METHOD OF JET VIBRATION.

889

TABLE of log /x. (x)+ 10.
= 2, 3, 4, 6. O^x^

X.

Log/*,.

LogP*.

Log m.

LogfH.

0-00

10-07248

9-47042

9-07248

8-52842

o-oi

0-02
0-03
0-04

10-07247
10-07241
10-07232
10-07219

9-47042
9-47039
9-47036
9-47031

9-07248
9-07247
9-07245
9-07242

8-52841
8-52841
8-52840
8-52839-

0-05

10-07203

9-47024

9-07238

8-52837

0-06
0-07
0-08
0-09

10-07183
10-07160
10-07133
10-07102

9-47016
9-47007
9-46996
9-46984

9-07234
9-07229
9-07223
9-07216

8-52835
8-52833
8-52830
8-52827

o-io

10-07068

9-46970

9-07209

8-52824

0-11
0-12
0-13
0-14

10-07030
10-06988
10-06943
10-06895

9-46955
9-46988

9-46920
9-46901

9-07200
9-07191
9-07181
9-07170

8-52820
8-52816
8-52812
8-52807

0-15

10-06843

9-46880

9-07159

8-52802

0-16
0-17
0-18
0-19

10-06787
10-06728
10-06665
10-06599

9-46857
9-46834
9 • 46808
9-46782

9-07147
9-07133
9-07119
9-07105

8-52797
8-52791
8-52785
8-52778

0-20

10/06529

9-46753

9-07089

8-52771

0-21
0-22
0-23
0-24

10-06455
10-06379
10-06298
10-06215

9-46724
9-46693
9-46661
9-46627

9-07073
9-07056
9-07038
9-07019

8-52764
8-52757
8-52749
8-52740

0-25

10-06128

9-46592

9-07000

8-52732

0-26
0-27
0-28
0-29

10-06037
10-05943
10-05846
10-05745

9-46555
9-46517
9-46477
9-46436

9-06980
9-06959
9-06937
9-06914

8-52723
8-52714
8-52704
8 52694

390

MR. P. 0. PEDERSEN ON THE SURFACE-TENSION OF LIQUIDS

TABLE of log /i» (x)+ 10 (continued).

X.

Log /io.

Log/tj.

Log /*4.

Log /i«.

0-30

10-05641

•

9-46394

9-06891

8-52684

0-31
0-32
0-33
' 0-34

10-05533
10-05422
10-05308
10-05191

9-46350
9-46305
9-46259
9-46211

9-06867
9-06842
9-06816
9-06790

8-52673
8-52662
8-52650
8-52639

0-35

10-05070

9-46162

9-06762

8-52627

0-36
0-37
0-38
0-39

10-04946
10-04819
10-04689
10-04555

9-46111
9-46059
9-46005
9-45951

9-06734
9-06706
9-06676
9-06646

8-52614
8-52601
8-52588
8-52575

0-40

10-04418

9-45894

9-06614

8-52561

0:41
0-42
0-43
0-44

10-04279
10-04136
10-03989
10-03840

9-45837
9-45778
9-45717
9-45656

9-06582
9-06550
9-06516
9-06482

8-52547
8-52532
8-52517
8-52502

0-45

10-03688

9-45593

9-06447

8-52487

0-46
0-47
0-48
0-49

10-03533
10-03374
10-03213
10-03049

9-45528
9-45463
9-45396
9-45327

9-06411
9-06375
9-06337
9-06299

8-52471
8-52454
8-52438
8-52421

0-50

10-02881

9-45257

9-06260 _

8-52404

0-51
0-52
0-53
0-54

10-02711.
10-02538
10-02362
10'02183

9-45186
9-45114
9-45040
9-44965

9-06221
9-06181
9-06139
9-06098

8-52386
8-52368
8 • 52350
8-52331

0-55

10-02002

9-44889

9-06055

8-52312

0-56
0-57
0-58
0-59

10-01817
10-01630
• 10-01440
10-01248

9-44811
9-44732
9-44652
9-44570

9-06011
9-05967
9-05922
9-05877

8-52293
8-52273
8-52253
8-52232

INVESTIGATED BY THK METHOD OF JET VIBRATION.

391

TABLE of log fin(x)+lO (continued).

z.

Log/*,.

Log /*s.

Login.

Log/^

0-60

10-01052

9-44487

9-05830

8-52212

0-61
0-62
0-63
0-64

10-00854
10-00654
10-00450
10-00245

9-44403
9-44317
9-44231
9-44143

9-05783
9-05735
9-05687
9-05637

8-52191
8-52169
8-52147
8-52125

0-65

10-00036

9-44053

9-05587

8-52103

0-66
0-67
0-68
0-69

9-99825
9-99612
9-99396
9-99178

9-43963
9-43871
9-43778
9-43684

9-05536
9-05485
9-05433
9-05379

8-52080
8-52057
8-52034
8-52010

0-70

9-98957

9-43688

9-05326

8-61986

0-71
0-72
0-73
0-74

9-98734
9-98508
9-98280
9-98050

9-43491
9-43393
9-43294
9-43194

9-05271
9-05216
9-05160
9-05103

8-51961
8-51936
8-51911
8-51886

0-75

9-97818

9-43092

9-05046

8-51860

0-76
0-77
0-78
0-79

9-97583
9-97346
9-97107
9-96866

9-42989
9-42885
9-42780
9-42674

9-04988
9-04929
9-04869
9-04809

8-51834
8-51807
8-51781
8-51753

0-80

9-96622

9-42566

9-04748

8-51726

0-81
0-82
0-83
0-84

9-96377
9-96129
9-95880
9-95628

9-42457
9-42347
9-42236
9-42124

9-04686
9-04624
9-04561
9-04497

8-51698
8-61670
8-51641
8-51613

0-85

9-95374

9-42010

9-04432

8-51583

0-86
0-87
0-88
0-89

9-95118
9-94861
9-94601
9-94340

9-41896
9-41780
9-41663
9-41545

9-04367
9-04301
9-04235
9-04167

8-51554
8-51524
8-51494
8-51463

392 MR. P. 0. PEDERSEN ON THE SURFACE TENSION OF LIQUIDS, ETC.

TABLE of log ftB(a;) + 10 (continued).

X,

Logfr.

Log /is.

Log /*4.

Log/j«.

0-90

9-94076

9-41426

9-04099

8-51433

0-91
0-92
0-93
0-94

9-93811
9-93544
9-93275
9-93005

9-41306
9-41185
9-41062
9-40939

9-04031
9-03961
9-0389'!
9-03821

8-51402
8-51370
8-51338
8-51306

0-95

9-92732

9-40814

' 9-03749

8-51274

0-96
0-97
0-98
0-99

9-92458
9-92183
9-91905
9-91626

9-40689
9-40562
9 • 40434
9-40305

9-03677
9-03604
9-03531
9-03457

8-51241
8-51208
8-51174
8-51141

1-00

9-91346

9-40175

9-03382 .

8-51106

[ 393 1

X. The Normal Weston Cadmium Cell

By F. E. SMITH, A.R.C.Sc.
(From the National Physical Laboratory.)

Communicated by R. T. GLAZEBROOK, F.R.S.

Received July 13, — Read November 21, 1907.

[PLATE 5.]

THE experimental investigations described in this communication had as their
primary object the improvement of the Clark and Weston Cadmium Cells as
standards of electromotive force.

The older investigations of RAYLEIGH,* KAHLE.t and GLAZEBKOOK and SKINNER^
proved the Clark cell to be very trustworthy, and only within the last few years has
any serious attempt been made to displace it from the premier position in which it
was placed in 1894. In 1892 WESTON§ introduced the cell bearing his name. This
cell contains a solution of cadmium sulphate instead of zinc sulphate, as in the Clark,
and an alloy of cadmium and mercury forms the negative pole. ' As originally
specified, the solution was saturated at 4° C., and no crystals of cadmium sulphate
were inserted in the cell ; under normal conditions there was therefore no change in
the concentration for small variations in temperature. When the solution is saturated
at all temperatures, i.e., when solid cadmium sulphate is always present in the cell,
the name " Cadmium Cell " has been frequently assigned to it in order to distinguish
it from the original form. In this communication the latter cell is the type experi-
mented with, and since it is referred to as the Westou Cadmium Cell in the reports
of the International Conference the same name has been adopted by the author.

* Lord RAYUUGH, ' Phil. Trans.,' 175, p. 412, 1884, and 176, p. 781, 1886.
t K. K.UII.K, 'Zeitech. f. Instnimentenk.,' 12, p. 117, 1892, and 13, p. 293, 1893.
; R. T. GLAZBBROOK and S. SKINNER, ' Phil. Trans.,' 183, p. 567, 1892.
§ WKSTON, ' The Electrician,' voL 30, p. 741, 1892.
VOL. CCV1I.— A 422. 3E 16.1.08

394 MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

Lord RAYLEIGH* was the first to point out that the quality of the mercurous
sulphate used as the depolariser in Clark cells was a cause of variation in E.M.F. ;
and Mr. SwiNBURNEf arrived at the same conclusion in 1891. Later JAEGER and
LiNDECKj attributed similar variations in the Weston cadmium cell to the same salt.

In 1902 some experiments at the National Physical Laboratory§ plainly indicated
that the mercurous sulphate might produce variations in the E.M.F. as great as
0'002 volt, or 1 part in 700 of the voltage of the Clark cell. A new specification of
the mode of manufacture of the depolariser was thought to be desirable, and
experiments were immediately made with this end in view.

Almost simultaneously CARHART and HULETT,|| at the University of Michigan, and
WoLFF,1T at the National Bureau of Standards, Washington, attacked the same
problem, and a little later H. v. STEINWEHR,*^ at the Physikalisch-Technische
Reichsanstalt, made a special study of the change of E.M.F. produced by varying the
size of the crystals of the depolariser. While it must be admitted that the chemistry
of the standard cell is still incomplete, an analysis of results shows that different
observers can set up cells of almost identical E.M.F., and that their constancy is many
times that of the standards used ten years ago.

PREPARATION OF THE MATERIAL.
Mercury.

In all our work the commercial mercury was cleaned with dilute nitric acid, washed
with distilled water, and distilled twice in vacuo.

The Depolariser.

At the National Physical Laboratory the mercurous sulphate has been prepared in
four ways: (1) Electrolytically ; (2) by chemical precipitation, mercurous nitrate
being added to sulphuric acid ; (3) by the re-crystallisation of purchased samples of
mercurous sulphate from strong sulphuric acid ; and (4) by the action of fuming
sulphuric acid on mercury.

The first of these methods was developed in 1904 by CARHART and HULETT|| and
also independently by WOLFF. If The mercurous sulphate is formed at a mercury

* Lord RAYLEIGH, ' Phil. Trans.,' 175, p. 412, 1884, and 176, p. 781, 1886.
t J. SWINBURNE, ' British Association Report,' Section A, 1891.
t W. JAEGER and ST. LINDECK, ' Zeitschr. f. Instrumentenk.,' 21, p. 33, 1901.
§ F. E. SMITH, ' British Association Report,' Section A, 1904.

|| H. S. CARHART and G. A. HULETT, ' Amer. Electrochem. Soc. Trans.,' 6, pp. 109-126, 1904.
H F. A. WOLFF, 'Amer. Electrochem. Soc. Trans.,' pp. 49-58, April 7, 1904.

** H. v. STEINWEHR, ' Zeitschr. f. Instrumentenk.,' 25, pp. 205-208, July, 1905 ; also ' Zeitschr. f. Elek-
trochem.,' pp. 578-581, 190G.

\IK I. K SMITH ON THE NORMAL WK.STOX CADMIUM CELL. 395

anode in dilute sulphuric acid, the latter, in the experiments made at the National
Physical Laboratory, consisting of 1 volume of strong sulphuric acid (density 1'84) to
5 volumes of water ; its strength was, therefore, about 3'0 molecular. The anode
surface was kept well exposed by a glass stirrer, and the current density was from

1 to 5 amperes per 100 sq. centims. of anode surface. In the second method
purchased protonitrate of mercury was sometimes used, but more often it was made
from mercury and nitric acid. About 1 5 cub. centims. of concentrated nitric acid
was added to 100 grammes of mercury, and when the action was over, or nearly over,
the resulting solution was added to 200 cub. centims. of dilute nitric acid (1 of acid
to 40 of water). The acid solution of mercurous nitrate thus formed was run as a
very fine stream from the narrow orifice of a pipette into 1000 cub. centims. of hot
dilute sulphuric acid (1 to 3), the liquid being well stirred during the mixing.
Mercurous sulphate was precipitated. It was washed two or three times by
decantation with dilute sulphuric acid (1 to 6) and filtered. The third method of
manufacture is more costly. A purchased sample of the salt is heated with
concentrated sulphuric acid to a temperature of about 150° C. and the hot clear acid
carefully poured into dilute sulphuric acid (1 to 6), when precipitation of pure
mercurous sulphate results. The fourth method, as originally employed, is trouble-
some. Fuming sulphuric acid is added to pure distilled mercury and stirred well
until the action between the two is practically at an end. Mercurous sulphate is
thus formed in the cold and appears in the crystalline form after a few minutes.
Equally satisfactory results are obtained, however, if sufficient mercury is placed in a
crystallising dish to cover the base and the fuming sulphuric acid added to a depth of

2 or 3 millims. The dish is covered with a clock glass and placed in a dark room for
one or two weeks.

In all the methods of production the resulting mercurous sulphate was washed two
or three times by decantation with dilute sulphuric acid (1 to 6) and afterwards
introduced into a Buchener funnel for the removal of the acid as completely as
possible by exhaustion with a filter pump. The sides of the funnel were washed
down with neutral saturated cadmium sulphate solution and the salt washed
5 or 6 times with more of the same solution. About 5 cub. centims. was needed
for each washing. In a few instances the sulphate was straightway employed for
the manufacture of the depolarising paste, but in the majority of cases it was
transferred together with a little of the cadmium sulphate solution to a small stock
bottle. After a week or ten days the solution was always slightly acid to congo red
ji;i[KM' and the mercurous sulphate was therefore washed once more before using.

In much of the earlier work absolute alcohol which had been specially distilled was
employed for washing the mercurous sulphate, and the salt was stored in contact with
more of the same liquid. We cannot, however, recommend this procedure, as we
l>elieve slight hydrolysis results, owing to the absorption of moisture by the
alcohol.

3 E 2

396 MR. P. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

Cadmium Amalgam.

The amalgam has been prepared in two ways, (1) by depositing cadmium
electrolytically in a weighed quantity of pure mercury, the electrolyte being an
acid solution of cadmium sulphate, (2) by heating together pure cadmium and
mercury, the resulting mass being washed with dilute sulphuric acid to remove the
surface dross. The amalgams used have been one part of cadmium to seven parts of
mercury (12^ per cent.) or one part of cadmium to nine parts of mercury (10 per cent.).
We have employed amalgams of other concentrations, but not in the cells discussed in
this paper. The investigations of DEARLOVE,* KEEP and BOETTGER,! BUL^
PUSCHIN,§ and JAEGER|| have shown that attention must be paid to the percentage of
cadmium m the amalgam. It appears probable that in an amalgam containing
between 6 and 14 per cent, of cadmium, there is at normal temperatures a solid and
a liquid phase, of which the former is an isomorphous mixture of mercury and
cadmium. The E.M.F. of the amalgam towards a cadmium sulphate solution does
not depend on the relative amounts of the two phases, and on adding cadmium to the
amalgam no change in E.M.F. occurs therefore until the liquid phase disappears.
Similarly, if cadmium is extracted from the mixture the E.M.F. is constant until the
solid phase disappears. A rise in temperature increases the liquid and diminishes
the solid phase : for any particular amalgam there are, therefore, limits of tempera-
ture between which the two phases are always present. JAEGER has shown that for
all amalgams in which the two phases exist the E.M.F. towards a cadmium sulphate
solution is constant for a given temperature. DEARLOVE first proposed a 12^ per cent,
amalgam ; this is satisfactory at all ordinary temperatures and has been generally
employed.

The Cadmium Sulphate.

We have usually ground the purchased crystals and made a saturated or nearly
saturated solution by agitation and warmth. This was filtered to clear and placed in
crystallising dishes to slowly evaporate. The resulting crystals were well washed
several times with water and the final solution tested for acidity with congo red
paper.

A saturated solution of cadmium sulphate yields crystals of the composition
CdSO4|H2O at all temperatures up to 74°C., when CdSO4H2O separates instead.
KOHNSTAMM and COHEN! believed that they had discovered a transition point at

* A. DEARI.OVE, 'The Electrician,' vol. 31, p. 645, 1893.

t KKRP and BOETTGER, 'Zeitschr. f. anorgan. Chem.,' 25, p. 1, 1900.

J BIJL, 'Zeitschr. f. Phys. Chem.,' 41, p. 641, 1900.

§ PUSCHIN, ' Zeitschr. f. Phys. Chem.,' 34, p. 621, 1901.

|| W. JAF.GEK, 'WiED. Ann.,' 65, p. 106, 1898; 'Zeitschr. f. Instrumentenk.,' 20, p. 317, 1900.

f T. KOHNSTAMM and E. COHEN, ' WIED. Ann.,' 65, p. 344, 1898.

Mfc F. E. SMITH OX THE NORMAL WESTOX CADMIUM CKLL. 3y?

•

about 17°C., but the irregularities were afterwards traced by COHEN* to a transition
which the 14'3 per cent, amalgam which was used undergoes at 23° C. The solution
has also been investigated by H. v. STEINWEHB,! who failed to confirm any transition
point at about 17° C. Cadmium sulphate is very soluble and increases very little in
solubility over the ordinary range of temperature. As purchased, the crystals are
generally acid, and in all cases it appeal's necessary to purify by recrystallisation.

Setting up of the Cell.

We have employed the Rayleigh H form of cell in nearly all our work. A
platinum wire was fused into the lower end of each limb, and the parts of the wire
inside the vessel were amalgamated by passing an electric current from a platinum
anode through an acid solution of mercurous nitrate to each of the wires in turn.
The vessel was washed out twice with dilute nitric acid, and several times with
distilled water ; it was dried in an oven. A small pipette was used for the intro-
duction of the amalgam, and a small thistle funnel for the insertion of the mercurous
sulphate paste and cadmium sulphate crystals. The main stock of amalgam was
flooded with very dilute sulphuric acid, and melted over a water bath ; a little was
then introduced into one of the limbs of the H vessel. After the amalgam had
solidified the limb containing it was washed out several times with distilled water,
care being taken not to wet the interior of the other limb. A little distilled water
was then added, and the amalgam again melted by immersing the H vessel in
hot water ; after solidification it was washed once more. Into the other limb
of the vessel sufficient mercury was added to cover the amalgamated platinum
wire and then the mercurous sulphate paste was introduced. The paste consisted
of mercurous sulphate mixed with about one-fourth its volume of powdered
recry stall ised cadmium sulphate, and about one tenth its volume of pure mercury.
(The latter was not added when the mercurous sulphate was prepared electrolytically
or by means of fuming sulphuric acid.) To this mixture sufficient saturated cadmium
sulphate solution was added, so that when well mixed the whole formed a thin paste.
After the introduction of some of this paste into the limb containing the mercury,
powdered crystals of cadmium sulphate were added to the contents of each limb, and
after an interval of one hour sufficient saturated cadmium sulphate solution was
inserted to fill the vessel to the top of the cross-connecting-tube. The cells were
hermetically sealed with the aid of a blow-pipe. For the comparison of their electro-
motive forces the cells were immersed in paraffin oil and were maintained at an
approximately constant temperature of 17° C. The comparisons were made by means

* E. COHEN, ' Zoitschr. f. Phys. Chem.,' 34, p. 621, 1901. W. JAEGER and ST. LINDECK, ' Ann.
d. Physik (4),' 3, p. 366, 1900; also ' Zeitschr. f. Phys. Chem.,' 35, p. 98, 1900.
t H. v. STK.IMVEJIR, 'Ann. d. Physik, pp. 1046-1053, 1902.

398 MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

of a high-resistance (15,000 ohms) potentiometer made by O. WOLFF, of Berlin, nml
a Broca galvanometer of 1000 ohms resistance made in the workshops of the National
Physical Laboratory. It was quite easy to read to one hundred-thousandth of a volt,
and, if necessary, one-tenth of this could be estimated with considerable accuracy.

When testing samples of mercurous sulphate we have often used a four-limb vessel
similar to two Kayleigh H-form vessels crossed. Cadmium amalgam was placed in
one limb, and in the other three mercury and the depolarisers were inserted. The
electrolyte was a saturated solution of cadmium sulphate.

Unit of Electromotive Force.

In a recent communication to the Royal Society, Professor AYRTON, Mr. MATHER,
and the author* have given the E.M.F. of some of the cells included in Table I. in
terms of the ampere (10"1 C.G.S.) and the international ohm. While it must be
admitted that this E.M.F. is possibly different from the true E.M.F. in volts
(10~8 C.G.S.) by 2 or perhaps 3 parts in 10,000, it is probably the most accurate
value known, and has the further advantage of being the mean of values the
observations of which extended over 19 months. A further deduction is that the
E.M.F. of most of the cells under observation did not change in this period by more
than O'OOOOl volt. We have therefore given the E.M.F. of cells in Table I. in terms
of the ampere (10"1 C.G.S.) and the international ohm.

It is impossible to give all the observations over the period 1904-1907 ; those
given are at approximately equal intervals of time. In cases where considerable
changes in the E.M.F. have resulted, more extensive observations are given in
subsequent tables. -The values for the period May, 1904, to October, 1905, have
been deduced from intercomparisons of cells, as no value in terms of tl>e ampere and
international ohm could be assigned until the later date.

Table I. gives the results of the observations on 60 cells. Since 1904 more than
200 cells have been set up, and a few of those which seem to be most valuable from
the point of view of results obtained are included in the table. For the 60 cells in
Table I., 16 samples of mercurous sulphate have been used, 6 of cadmium amalgam
and 5 of cadmium sulphate. In some cases the mercurous sulphate was washed with
alcohol ; the letter A is then inserted in column 5 of the table. The approximate
depth of the paste is given in column 6, and the numbers in column 7 indicate the
range of the dimensions of the mercurous sulphate crystals in thousandths of a
millimetre. Particulars of the cadmium amalgam are given in columns 8 and !) ;
E indicates that the amalgam was prepared by the electro-deposition of cadmium,
and F by the fusion of cadmium and mercury. The mercurous sulphate was usually
prepared two weeks in advance of its use as a depolariser in a cell.

* W. E. ATRTON, T. MATHEK, and F. E. 8111x5, "On a New Current Weigher," 'Phil. Trans.,' 1907.

MR. F. K SMITH ON T!!K X<>KM.\L WESTON CADMIUM (I.I I

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400

MR. F. E. SMITH ON THE NORMAL WESTON CADMII'M C'KI.I,

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MR. F. E. SMITH ON THE NuKMAL WESTON CADMIUM CELL. 401

DISCUSSION OF THE RESULTS.

I '•"' • ' "f Various Amalgams.

The E. M.F.'s of the cells set up with amalgams prepared at different times agree
under otherwise equal conditions within 1 part in 100,000. The amalgam prepared
by the electrodeposition of cadmium in mercury is perhaps preferable for the cells of
standardising institutions, but our observations do not show any certain difference
between it and that prepared by the fusion of cadmium and mercury, when the latter
amalgam is freed from dross.

Effect of the Cadmium Sulphate.

We have obtained our cadmium sulphate from various sources, but after one
crystallisation and thorough washing of the crystals with water no certain difference
in the behaviour of the solutions has been detected.

Effect of the Depolarizer.

The mean value of the cells set up with the electrolytic mercurous sulphate is
T01828 volts ; that of the cells containing the salt prepared by chemical precipitation
(Method II.) is 1*01830 volts ; when mercurous sulphate was employed which had
been precipitated from hot strong sulphuric acid, TO 1832 volts is the mean ; and the
salt prepared with fuming sulphuric acid gives 1 '01831 volts.

We conclude that the mode of manufacture of the mercurous sulphate is immaterial,
provided that certain conditions are observed, and our guiding principle in the
manufacture of the salt and the preparation of the paste is to prevent hydrolysis
by keeping the salt in contact with dilute sulphuric acid (1 to 6), or with saturated
cadmium sulphate solution. This is in accordance with HULETT'S investigations.

CARHART and HULETT* have examined Weston cadmium cells containing electro-
lytic and chemically prepared sulphate (Method II.), and conclude that there is no
appreciable difference in E.M.F. Later, HuLErrt constructed two other cells
containing the chemical sulphate, and found- them about O'OOOIS volt higher than
cells containing the electrolytic salt. He concludes that the electrolytic sulphate is
the most reliable preparation. Dr. F. A. WOLFF and C. E. WATERS^ have examined
many more specimens, and conclude that the four methods dealt with in this
communication give practically identical results. They have also examined samples
of mercurous sulphate prepared by the action of sulphuric acid containing a small
percentage of nitric acid on mercury (Lunge reaction) ; by the reduction of mercuric
sulphate by mercury, and by digesting commercially pure samples of mercurous

* H. S. CARHART and U. A. HULETT, ' Amer. Electrochem. Soc. Trans.,' 6, pp. 109-126, 1904.
t G. A. HULETT, 'Phys. Rev.,' 22, pp. 321-338, June, 1906.

» F. A. WOLFF and C. E. WATERS, 'Electrical World,' 49, pp. 100, 101, January 12, 1907.
VOL. COVII. — A. 3 F

402

MR F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

uulphate with sulphuric acid. The last method gave the largest difference, but this
was of the order of 5 parts in 100,000 only.

Mr. J. A. SADD, A.C.G.I., a student demonstrator of the Central Technical College,
has constructed some cadmium cells in accordance with a specification published by
the author* in 1905. These cells contain mercurous sulphate prepared electrolytically,
and were forwarded to the National Physical Laboratory for comparison with our
standards. There are ten cells in all, and their values at 17° C. are as follows : —

1-01832 volts.
34
35
37
38

1-01839 volts.
39
40
42
43

Mr. H. TINSLEY, of Beckenham, Kent, has also prepared some cadmium cells in
accordance with the specification mentioned above. In this case the depolariser was
prepared with mercurous sulphate precipitated by adding mercurous nitrate to
sulphuric acid (Method II.). Most of Mr. TINSLEY'S cells are also greater in E.M.F.
than the N.P.L. standards, the difference being about 1 part in 10,000.

In May, 1907, 12 cadmium cells prepared by Dr. F. A. WOLFF of the National
Bureau of Standards, Washington, were brought to England by Dr. BURGESS, and a
direct comparison between the cells of the two institutions was thus rendered possible.
Cells which were believed to nearly represent the normal cell were chosen to compare
with those from Washington, and the following differences were observed : —

TABLE II.

National Bureau of Standards

National Physical Laboratory

cadmium cells.

cadmium cells.

E.M.F. of cell minus

RM.F. of cell minus

Cell.

mean E.M.F. of all

Cell.

mean E.M.F. of all

microvolts.

microvolts.

WP 8

+ 1

P 52

+ 2

9

_ Y

„ 53

-1

10

0

„ 54

-1

W 19

+ 1

„ 55

-1

105

-11

„ 210

+ 2

181

0

C 12

-2

1S2

0

„ 17

+ 3

183

+ 2

„ 19

0

184

+ 1

„ 117

-1

185

+ 6

H 26

+ 2

, 18Q

1

„ 28

+ 4

, 18V

+ 4

„ 29

-2

F. E. SMITH, 'British Association Rt-port,' Section A, 1905

MR. F. K. SMITH ON THE NORMAL WFSTON CADMIUM CELL. 403

The mean E.M.F. of the 12 " National Bureau of Standards" cells is less than the
mean E.M.F. of the 12 N.P.L. cells by 3 microvolts.

P 52, P 53, P 54, and P 55 were set up in November, 1906,
C 12, C 17, C 19, and C 117 „ „ June

H 26, H 28, and H 29 „ „ February, 1907,

P210 „ „ March

The mean E.M.F. of the 12 N.P.L. cells is not quite the mean of all the N.P.L.
cadmium cells, which are believed to be normal ; the latter cells have the higher mean
by about 2 parts in 100,000.

Eleven of the twelve cadmium cells from Washington contain electrolytic mercurous
sulphate; the twelfth, W 105, contains mercurous sulphate prepared by the Lunge
reaction. The depth of the pastes in these cells is about 1'5 centims. The twelve cells
of the National Physical Laboratory contain mercurous sulphate prepared chemically
(Method II.). and the depth of the paste is about 0'5 centim. The nett result of
these comparisons with other observers is that mercurous sulphate of sufficiently
uniform properties can be prepared in several ways, provided that certain conditions
are observed. The possibility of an approximately constant size of mercurous sulphate
crystal resulting from all the methods must not, however, be overlooked, and as the
size of the crystal has not been stipulated by us, it is necessary to examine the
evidence on this point.

H. v. STEINWTSHK* was the first to call attention to the part played by the size
of the crystal, and our method of investigation is very similar to that employed by him.

Effect of the Size of the Crystals of Mercurous Sulphate.

Twenty samples of mercurous sulphate have been examined under the microscope,
and in twelve cases microphotographs have been taken, the magnification being 250.
Of these, eleven are reproduced on Plate 5. It will be seen from these micro-
photographs that not only do the crystals vary considerably in size, but that each
method of preparation produces its own particular type of crystal. The electrolytic
method, as employed by us, gives small crystals, somewhat imperfect in shape, and
varying in size from 2 to 8 microns (0'002 to O'OOS millim.). The crystals resulting
from the chemical precipitation method are very much rounded, like pebbles ; they
have the appearance of being formed from perfect crystals, the edges of which have
been rounded by the solvent action of the hot dilute sulphuric acid. The uniformity
in the size of the crystals is more marked in the samples produced by this method
than in the specimens made in any other way tried by us. The size of most of the
crystals in figs. 2 to 4 varies from 5 to 30 microns. The mercurous sulphate produced

* H. v. STKINWEHB, « Zeitschr. f. Instrumentenk.,' 25, pp. 205-208, July, 1905.

3 F 2

404

MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

by the action of fuming sulphuric acid on mercury is evidently in the form of large
crystals, but most of these appear to get broken in the washing processes, with the
result that the size of the grain is considerably diminished. Figs. 9 and 10 show
these fragments of crystals, which vary in size from 5 to 25 microns.

We have not taken any microphotographs of crystals produced by Method III., as
this method is only of interest as an independent means of obtaining mercurous
sulphate ; it is far too troublesome to be generally employed. We have, however,
prepared some very large crystals by a method suggested by H. v. STEIN WEHR.* A
dilute acid solution of mercurous nitrate was allowed to flow very slowly indeed from
a burette into dilute sulphuric acid (1 to 4) heated to 100° C. over a water-bath.
Small crystals of mercurous sulphate were at first formed, which grew, however, in
the slightly super-saturated solution of mercurous sulphate until many of them
attained a length of a few millimetres. The crystals were washed with dilute
sulphuric acid (1 to 6) and the finer particles were removed by agitation with the
acid and rapid decantation. The resulting product was undoubtedly of large grain,
and there were numerous particles present of the size shown in fig. 5. The length of
this crystal is about 400 microns.

In September, 1906, three cells were set up which contained these large crystals
of mercurous sulphate as a depolariser. Table III. gives the results of observations
on these cells from October, 1906, to May, 1907.

TABLE III.

0*

1906.

1907.

October.

November.

December.

January.

February.

March.

April.

May.

SI
S2
S3

1-01843
36
40

42
35
38

42
35
39

40
34
38

41
35
40

41
34
38

40
34
39

39
35
38

The cells are not in very good agreement and the mean E.M.F. is about 8 in 100,000
higher than the E.M.F. of the -normal cells in Table I. Recently we have set up
more cells with a second sample of large-grained mercurous sulphate, and they also
have E.M.F.'s higher than T01830 volts. In order that there should be no doubt
about the size of the crystals in S 1, S 2, and S 3, they were unsealed in June, 1907,
the pastes washed with dilute sulphuric acid to remove the cadmium sulphate crystals
and the residue examined microscopically. It was apparent that during the prepara-
tion of the depolarising paste and its insertion in the cell many of the large crystals

* H. V. STEINWEHR, ' Zeitschr. f. Instrumentenk.,' 25, pp. 205-208, July, 1905 ; also ' Zeitschr. f,
Electrochem.,' pp. 578-581, 1906.

MR. F. E. SMITH ON THK NORMAL \VKSTON CADMIUM CELL.

40D

wore broken, with the result that the mean size of the crystals was diminished, hut
the fragments were still much larger than the crystals prepared by any of the other
methods. Figs. 1 and 8 (Plate 5) are microphotographs of some of the crystals of
mercurous sulphate after their removal from the cells Si, S 2, and S 3. It was
olwerved that in all cases there was a small number of comparatively small crystals,
or fragments of crystals, associated with the large ones, but we found this unavoidable.
In one instance we took very great pains to eliminate small particles, and succeeded
in doing so to a considerable extent, but subsequent examination of the crystals after
the manufacture of the depolarising paste and its insertion in a cell showed that their
size had been appreciably reduced.

T.I hie IV. enables a comparison to lx> made Iwtween cells set up with crystals of
various sizes.

TABLE IV.

Cells.

Mercurous sulphate.

Size of most of the crystals
in microns.

Meau E.M.F.

E 62, &c.

Electrolytic I.

5 to 15

1-01828

E 80,

I-

3 15

28.-,

HA 1,

Chemical II.

5 30

30,

B 41,

„ II.

5 30

29

HA 7,

„ II.

5 30

30

P 410,

» II.

5 30

30

N 23,

Fuming acid IV.

5 20

31

SI, S 2, S3

STEINWEHR'S method «/

about 10 per cent., 20
„ 30 „ 60
„ 40 „ 100
aljout 20 per cent., greater than 100

I 37

The higher E.M.F. of the cells S 1, S 2, S 3 is probably due to slight impurities in
the mercurous sulphate, and not to the large size of the crystals. During the
preparation of these crystals the sulphuric acid was not stirred, and the only agitation
of the liquid was that produced by the mixing and by convection currents. It is
possible for slight hydrolysis to result under such circumstances, and also possible for
mercurous nitrate to be imprisoned in some of the large crystals which are formed.
The latter appears to be not improbable in our case.

If we exclude from Table IV. those cells set up with the very large crystals, we
have as the limits of the dimensions of the others 3 to 30 microns, and we conclude
that if the size of the crystals be within these limits they will have very nearly the
same solubility and practically give the same E.M.F. in a standard cell.

This conclusion is not in accordance with observations made by STEIXWEHR, who
first called attention to the possible effect of the size of the crystals, and claims that
it is a principal cause of the variations observed in standard cells. Lord KELVIN has
shown that the saturation pressure of small drops of water is greater than that of

406 MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

large drops; similarly the solubility of small particles is greater than that of large
ones, and therefore the saturated solution will show a greater E.M.F. than the saturated
solution of the large crystals. With fluctuating temperatures the tendency is for the
small crystals to disappear and the mean size of the crystals to increase ; this lowers
the concentration and with it the E.M.F. It is only for very small crystals that the
change in solubility is appreciable, and at present it is not possible to calculate the
change in E.M.F. produced by the variation in size.

STEINWEHR* examined several samples of purchased mercurous sulphate ; two such
suits from KAHLBAUM and MERCK respectively gave a difference in E.M.F. of 5 parts
in 10,000, and it was found that the dimensions of the crystals of the salt K giving
the higher E.M.F. were of the order of 1 micron, while the crystalline structure could
be seen in a number of particles of the other salt (M). The crystals of M were about
10 to 20 microns long. By grinding the M salt the difference was reduced to 1 or
2 parts in 10,000. Very large crystals were prepared by a method very similar to
that afterwards used by ourselves (p. 404), and the E.M.F. of a cell containing these
very large crystals was lower than that of a cell containing the K salt by 0'7 to
0'8 millivolt. By grinding some of the large crystals and setting up a cell with the
small particles as a depolariser the E.M.F. was increased by 0'6 millivolt, and was
therefore comparable with the K cell.

We also have set up cells with KAHLBAUM'S mercurous sulphate which was washed
with water and is therefore hydrolysed. The E.M.F. of these cells is at present
1'0186 volts and is constant ; indeed, from the point of view of constancy of E.M.F.,
these cells are as good as any of those dealt with in Table I. ; fig. 1 1 (Plate 5) is a
microphotograph of the salt used. The average size of the crystals is from 2 to
10 microns. The salt from KAHLBAUM, used by STEINWEHR, was smaller than this,
the particles being from 1 to 2 microns in length at the most. The difference of
0'3 millivolt found by us between cells set up with KAHLBAUM'S salt and those set
up with mercurous sulphate prepared by ourselves appears to be due not to a difference
in size of grain, but to the hydrolysis of the former salt.

HuLETrt has also measured the size of mercurous sulphate crystals prepared
electrolytically and found the particles to vary in length from 2 microns to 130 microns,
but has found no difference in the E.M.F. of cells set up with these crystals.

It is, of course, always possible that in an occasional preparation a very large
number of exceedingly fine crystals may be produced, and in such a case the change
of E.M.F. described by STEINWEHR will result, but unless there are numerous small
crystals in all of our preparations — and this is highly improbable — the large crystals
of mercurous sulphate which are sufficiently soluble to act as an efficient depolariser
cannot give an E.M.F. appreciably lower than those which are from 5 to 30 microns

* H. v. STEINWEHR, 'Zeitschr. f. Instrumentenk.,' 25, pp. 205-208, July, 1905; also 'Zeitschr. f.
Elektrochem.,' pp. 578-581, 1906.

t G. A. HUI.ETT, 'Phys. Rev.,' 22, pp. 321-338, June, 1906.

MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL. 407

in length. The question is an interesting one, and we hope to extend our investigation,
but at present there appears to be no necessity to stipulate the size of the crystals.
We would, however, draw attention to the uniformity of the size of the crystals of the
chemically precipitated salt, and as Method II. is the easiest of any of the methods of
preparation used by us, we strongly recommend it, or one founded on it, as a standard
method of preparation.

Constancy of the. Cell.

The results recorded in Table I. indicate remarkable constancy of most of the cells.
Eighty per cent, of the first fifty cells have apparently varied by not more than
2 parts in 100,000 since the first month of their preparation. Of the remainder, four
have varied from 3 to 5 parts, one 6 parts and one 16 parts in 100,000. There are
four others which belong to the N group of cells ; these were low at first, but rose to
the normal value 3 months after preparation. Further remarks on these four cells
appear on p. 410. The last eight cells in Table I. are anomalous. In 12 months the
three M cells have fallen 10, 16 and 21 parts in 100,000 respectively, and the five
D cells have values about 30 parts in 100,000 lower than the normal. Of the
approximately constant cells, twelve have been set up for 2^ years, twenty-three for
more than 18 months, and thirty-six for more than 12 months. It is important to
note that the comparatively new cells, i.e., those set up in March and April, 1907,
are practically identical in E.M.F. with the cells set up in 1904.

Of the abnormal cells it is noteworthy that, with the exception of E 67, they
belong to three groups and that the change in them has been practically continuous
from the time of their construction. The cells were prepared in a manner apparently
the same as that of the normal cells, and we are unable to offer a complete explanation
of their remarkable behaviour. Examination of the solutions shows that they are not
appreciably acid, and tests on the amalgams indicate that they give the same E.M.F.
as those used in the new cells. The depolarisers are, however, at fault. This we
have proved by connecting one of the cells with a new cell by means of a syphon and
measuring the E.M.F. between the negative pole of the new cell and the positive
I>ole of the abnormal cell. The same low E.M.F. was recorded. Something may have
happened during the preparation of the depolarisers for these abnormal cells, but we
are not aware of any departure from our usual proceeding.

There is, however, another indication that the depolarisers have changed, and we
wish to draw particular attention to this, as it may lead to the discovery of the cause of
the disturbance. In nearly all of the cells which have fallen by as much as 0'0002 volt
many of the small crystals of cadmium sulphate have changed in colour, and in some
cases the colour of the depolariser has also changed. The cadmium sulphate crystals
in these cells of low E.M.F. are of a yellowish- brown tint and the mercurous sulphate
in places is slightly yellow, or, in a few instances, green. This change has also been
noted in cells set up with some samples of purchased mercurous sulphate, but these

408 MR. F. E. SMITH ON THE NORMAL WESTOX CADMIUM CELL.

cells are high in E.M.F. and approximately constant. In the normal cadmium
cells the appearance of the mercurous sulphate is the same as when freshly prepared.
An interesting question is whether the change in colour of the paste is the result of
association with the other ingredients of the standard cell or not, and whether
a charging or discharging current produces or accelerates the change. The first part
of this question may be answered so far as association with the cadmium amalgam is
concerned, for we have kept many samples of the paste together with saturated
cadmium sulphate solution and mercury in stock bottles. In two of these pastes
green patches have appeared and the dimensions of the patches are slowly increasing.
The change is therefore independent of the presence of free cadmium or cadmium
amalgam. The probable acceleration of the change due to a charging or discharging
current is discussed on pp. 413-415.

A number of anomalous cells were closely examined in order to detect any other
peculiarities besides that of the coloration of the depolariser and crystals of cadmium
sulphate. In one of the cells a small bubble of air was imprisoned between the glass
and the paste, and the paste in contact with the bubble and to a depth of 1 millim.
from it was of a yellow tint, the remaining portion of the depolariser being normal in
colour. From these colour observations it appears probable that the change is of
the nature of a hydrolysis, but at present we have no reason to think that all the
pastes will, with time, similarly change, and that, as suggested by HULETT,* the
cathode leg of the cadmium cell is a system in unstable equilibrium. . Instead we are
inclined to believe that something abnormal occurred in the preparation of the
pastes.

Two cells which were low in E.M.F. by O'OOOl volt, but very constant, were opened,
and the solution in them was found to be very slightly acid. We regard this as an
indication of insufficient washing of the mercurous sulphate. Cell No. 2 was
employed in a potentiometer circuit during the estimation of current in absolute
measure by the Ayrton-Joues Balance. It was used from October, 1905, to June,
1907, and could not have fallen by more than 1 part in 100,000 during this period.
Its low initial E.M.F. is probably due to the presence of acid.

HULETT concludes that many of his cells have fallen 11 parts in 100,000 in a little
over two years, but that Clark cells have remained constant. In 1904 the author
pointed out that some cadmium cells made in 1902 had apparently fallen 0 '00007 volt ;
these, however, were prepared in the old way with purchased mercurous sulphate
washed with water, and since 1904 there is every reason to believe that they have
remained constant.

Drs. K. E. GTJTHE and C. L. v. ENDEt record the following results : — Three
cadmium cells were prepared by them on Nov. 2, 1906 ; by the 17th of the same
month each had fallen about 50 microvolts below the normal value ; by Dec. 13, 1906,

* G. A. HULETT, 'Phys. Rev.,' 23, pp. 166-183, August, 1906.

t K. E. GUTHE and C. L. v. ENDE, ' Phys. Rev.,' 24, pp. 214-221, February, 1907.

MR. F. E. SMITH ON THE NORMAL WESTON CAPMir.M CELL.

4U9

the mean fall was 100 microvolts, and there was evidence that they were still falling.
These three cells contained pastes prepared by Dr. GUTHE. On Nov. 10, 1906, four
other cadmium cells were prepared, and contained mercurous sulphate supplied by
Dr. HULETT. By Jan. 19, 1907, the mean fall of these cells was 0-00049 volt; on
the same date their average E.M.F. was about 0 '00023 volt lower than that of the
three cells previously considered.

Dr. GUTHE also gives the values of some Clark and cadmium cells set up with
electrolytic and chemically prepared pastes by Professors CARHART and HULETT. The
Clark cells appear to have remained constant since their construction in 1904, but
some of the cadmium cells have fallen by 3 to 4 parts in 10,000. In some cases more
than half of this change took place during the first year.

The results obtained at the National Physical Laboratory are, on the whole,
decidedly in favour of the constancy of the cell, and tend to show that the fall in
E.M.F. of certain N.P.L. cells is due to the mercurous sulphate in them being
somewhat abnormal when they were set up. An investigation of the pastes of
abnormal cells appears to be desirable, and may possibly lead to some explanation
of the want of constancy which the foregoing statement shows has been noted by
some observers.

E.M.F. of freshly prepared Cells.

When mercurous sulphate is freshly prepared and apparently free from acid, if
cells are set up with it as a depolariser on the same day as that of the precipitation
they do not usually take up their normal value immediately. They are sometimes
high at first, but fall rapidly in E.M.F., sometimes attaining their normal value
within a few hundred thousandths of a volt in a few hours, but more often an interval
of several days is required. An example of this is afforded in the case of B 151.
This was completed at 2.15 P.M. on the same day as the depolariser was manufactured ;
it was inserted in a constant temperature bath, and observations of the E.M.F. were
immediately made. At the same time another cell, B 149, was completed ; in this the
depolariser was mercurous sulphate whicli had been prepared three weeks previously
and had since remained in contact with saturated cadmium sulphate solution. This
cell attained its normal value almost immediately.

TABLE V.

Cell.

Day of preparation, March 7th, 1906.

March
8th.

March
9th.

2.20 P.M.

2.33.

2.37.

2.45.

3.0.

3.10.

3.30.

4.0.

5.0.

B151

1-01861

56

55

53

51

50

49

48

47

39

36

B149

1-01837

36

36

35

34

34

34

33

33

32

32

VOL. OCVH. — A.

3 O

410

MR. V. F, SMITH OX THK NORMAL WESTOX CADMIUM CELL.

" Ageing " of Cells.

Lord RAYLEIGH observed that the electromotive force of Clark cells when originally
set up was invariably high, and in some cases the fall in E.M.F. in a few weeks was
0'02 volt. This fall in E.M.F. immediately after manufacture has been confirmed by
numerous observers, and in consequence Clark cells were supposed to require
" ageing." The same is true of Weston cadmium cells if set up with pastes prepared
similarly to those used by Lord RAYLEIUH, but such extreme changes as 2 per cent,
have not come under the author's notice. The mercurous sulphate prepared by any
of the four methods described in this communication does not require " ageing," or to
a very small extent only.

Washing with Alcohol.

The group of cells, of which N 23 to N 26 are types, were abnormal in their
behaviour. The E.M.F. was at first 3 parts in 10,000 low, but gradually increased
until it was normal, and since then it has remained approximately constant. The
mercurous sulphate for these cells was washed with absolute alcohol to free the salt
from acid, but no attempt was made to remove the alcohol by further washing the
sulphate with saturated cadmium sulphate solution. The salt was removed from the
filter and immediately made into paste. Table VI. gives the observations from the

TABLE VI.

Cell.

Sept. 20,
1905.

Sept. 30.

Oct. 10.

Oct. 20.

Oct. 30.

Nov. 10.

Nov. 20.

Nov. 30.

Dec. 20.

Jan. 1,
1906.

N23

1-01800

06

11

19

22

23

25

27

29

32

,,24

03

10

14

18

23

25

27

28

30

33

,,25

00

04

08

12

17

21

24

26

32

32

,,26

10

13

16

19

22

25

28

29

31

32

,,17

37

35

33

33

33

32

33

32

32

32

date of preparation, September 20, 1905, to January 1, 1906; Cell N 17 contains
mercurous sulphate from the same sample, but which was freed from alcohol l>efore
making up the paste.

GUTHE and v. ENDE* prepared some mercurous sulphate which was not
thoroughly free from alcohol and found the E.M.F. of some Clark cells 0'00040 volt
lower than normal when set up with this, and there was no appreciable change in the
course of time. Their observations extended over threl§ months.

K. E. GITHE and C. L. v. ENDE, ' Phys. Rev.,' 24, pp. 214-221, February, 1907.

MK. F. E. SMITH ON THE NORMAL WE8TON CAI'MII'M CELL. 411

The Temperature Coefficient and " Lag."

In experimental work involving the use of the Clark cell, temperature corrections
have invariably to be introduced owing to the high value of the temperature
coefficient. This is the most serious objection to its use. The temperature coefficient
of the cadmium cell is much smaller and has been determined by JAEGER and KAHLE,
who give the following equation connecting temperature and E.M.F. :

E, = 1-0186-0-000038 («-20) -0'00000065 (t-

At the National Physical Laboratory six cells were chosen and their temperatures
were varied very slowly from 10° C. to 30° C. The maximum rate of change of
temperature was 1° C. per hour, and before making an observation at any particular
temperature the oil bath in which the cells were immersed was kept at that
temperature for at least an hour, a toluene thermostat capable of maintaining a
constant temperature to 0°'01 C. being employed. The cycle of temperature was
repeated three times. The agreement between the cells was excellent, and the mean
values of the E.M.F.'s were taken to obtain the temperature coefficient by the
method of least squares. The resulting temperature formula is

E, = E17-0-0000346(*-17) - 0 -00000066 («- 17)*.

This is in very good agreement with JAEGER and KAULE'S formula. The changes
in E.M.F. from 10° C. to 15° C., 10° C. to 20° C., and 10° C. to 30° C., as deduced
from the two formulae, are given below.

" C. JAKGER and KAHLE. N.P.L.

10 to 15 .' . . . ', 0-00015, volt. 0 -00014, volt.

10 „ 20 .;;-/ . .- 0-00032, „ 0-00031» „

10 „ 30 . -. ; . :-.--*- 0'000770 „ 0-00076, „

The lag of E.M.F. with respect to temperature changes was shown by AYRTON and
COOPER* to he much greater in the Board of Trade tube form of cell than in the
H form. They concluded that there is a " simple lag," which may be removed by a
comparatively short interval of constant temperature, and a " semi-permanent lag,"
which requires many hours of steady temperature for its complete removal. We have
made similar observations on the Weston cadmium cell and find evidence of the same
lag in it. The effect is, however, very small when the temperature changes are slow
and the range of temperature only a few degrees, as in the experiments of AYRTON
and COOPER on the Clark cell. When the range of temperature is about 1 5° C. and
the change of temperature very rapid, a difference in E.M.F. of about 30 microvolts is
often observed after the normal temperature of the cell has been restored for 4 or

* AYRTON and COUPKIE, ' Key. Soc. Proc-.,' 59, p. 368, 1896.
8 O 2

412

MR. F. E. SMITH ON THE NORMAL VVESTON CADMIUM CELL.

5 hours. Much, however, depends on the construction of the cell. An extreme
case is illustrated in fig. 12. Here a cell was maintained at a temperature of 55° C.
for 12 hours and was then plunged into a bath of paraffin oil at 17° C. The oil
was stirred, and observations of the E.M.F. were frequently made. At 1 1.20 A.M. (see

E.M.F.
TOI825

roiSOO

1-0177-5

TOI750

z

II- 20 a.m.

11-40

12-20 p.m.

12-40

12-0
TIM C

Fig. 12. Recovery curve of cell suddenly cooled from 55° C. to 17° C.

fig. 12) the cell was at a temperature of 55° C. and was then immersed in the oil at
17° C. Twenty minutes afterwards the E.M.F. was normal to 1 part in 4000; after
a total interval of 40 minutes it was right to 1 in 10,000, and after 1 hour to about
7 parts in 100,000. Fourteen days elapsed, however, before the cell was within
2 parts in 100,000.

Recovery after Short-circuiting.

In order to test the recuperative power of the Weston cadmium cell, one of the
cells was short-circuited for 1 minute, another for 5 minutes, a third for 5 hours,

MU. K. K. SMITH ON THE NORMAL WKSTON CADMIUM CELL.

413

and a fourth for 5 days. The recovery of the first two cells is illustrated in figs. 13
and 14. It will be observed that the cell which was short-circuited for 1 minute
was right within a ten-thousandth of a volt 1 minute afterwards, but 40 minutes
were occupied in its recovery to 1 part in 100,000. The cell which was short-

E M.F.

roiaso

I'OI825

I'OI820

roiais

O 5 IO (3

MINUTES
Fig. 13. Recovery curve of cell short-circuited for 1 minute.

circuited for 5 minutes was nearly 1 in 1000 low 1 minute afterwards ; at the end of
the second minute it was 1 in 2000 low, and after 5 minutes it had recovered within
1 in 5000 ; about l£ hours were required for its complete recovery. The restoration
of the E.M.F. of the third cell was much slower; 1 minute after the circuit was
opened its E.M.F. was about O'l volt, which value it appeared to retain for 3 minutes.
The E.M.F. then changed suddenly from O'l to 0'85 volt, and at the end of 4 minutes
its voltage was 0'9. The recovery was then more gradual. Ten miuutes after
breaking the circuit the E.M.F. was O'OOGl volt below normal, 20 minutes afterwards
0'0028 volt low, and 5 hours afterwards it was low by 0-00040. It recovered within
1 in 10,000 in 24 hours, but 3 weeks were occupied in its complete recovery.
The cell which was short-circuited for 5 days had an E.M.F. less than 0'05 volt
o minutes after breaking the circuit, and its E.M.F. did not rise above 0'08 volt for

414

MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

more than 6 hours. Twenty-four hours afterwards, however, its E.M.F. was normal
within 2 parts in 10,000, and it completely recovered within 6 weeks.

In 1884 Lord RAY LEIGH made some experiments on the polarisation of Clark cells
and conclusively showed that the effect of short-circuiting for a few minutes rapidly
passed away.

E.M.F.

roisso

I'OI780

TOI730

8

12

IS 2O

N/1 1 NOTES

24

28

32

Fig. 14. Recovery curve of cell short-circuited for 5 minutes.

Recovery curves for the Clark cell have been published by FISHER,* for a large
type of cadmium cell by TINSLEY,* and for the Calomel or Hibbert cell by HIBBEKT.*
An analysis of such curves leaves little doubt that short-circuiting is only temporarily
injurious.

Charging the Cell.

In practice, a standard cell is usually placed in a circuit in which a very small
current alternates in direction. These small charging and discharging currents can
have no immediate serious effect on the cell, as is amply proved by many cadmium
cells in use at the National Physical Laboratory which are frequently checked in the

* W. C. FlBHEK, " The Potentiometer and its Adjuncts," ' Electrician ' Series ; W. HIBBERT, ' The
Electrician,' vol. 37, p. 32, 1896; H. TINSLEY, "The Electrician,' vol. 47, p. 991, 1901.

MR. F. K. SMITH (IX Tin: NTORMAL WTvSTON CADMIUM CELL. 415

Electrical Standards Department. That a small current may be taken from the cell
without any permanent effect is proved from the olwervations when cells have been
sln>rt-niriiitf<l. hut the ell'ert <>f :i comparatively large charging current may be more
serious. Lord I! \YI.KHMI attempted to manufacture a Clark cell by the formation of
electrolytic mercurous sulphate inside an H vessel, the anode being mercury, the
electrolyte /.inc sulphate, and the cathode an amalgam of zinc. The cells so formed
were not constant, and their E.M.F.'s were low. In 1904 we attempted in a similar
way to produce cadmium cells, hut it was evident that normal mercurous sulphate was
not formed, as the resulting salt was highly coloured ; it was sometimes yellow, but
more often green. The fact that the depolariser in some of our anomalous cells has,
after a long period, turned a yellowish-green suggested to us that its formation might
he accelerated by small charging currents. The constancy of other cells subject to
the same treatment is certainly against such a view, hut a slight difference in the
original composition of the depolarisers might account for the more rapid change.
To test this point, we placed a normal cell in circuit with, but in opposition to, two
Leclanch6 cells for 18 hours. At the end of that time a green compound had formed
between the mercury electrode and the glass, but the depolariser appeared to be
unchanged. There is little doubt, however, but that some of the green salt was
present over the whole surface of the mercury. The E.M.F. of this cell was at first
very high, but in 4 weeks it gradually fell to 1 '01 833 volts. The observations which
we have so far made do not enable us to say whether any further fall is probable,
but it is evident that the small charging currents to which a cell is subjected in a
potentiometer circuit do not seriously affect its E.M.F.

Portability of tlie Cell.

Many of the cells made at the National Physical Laboratory are portable, and may
be sent through the post. In these cells the two limbs of the H .vessel are constricted
at points about 1^- centims. from their lower ends, and when making up the cell,
cadmium sulphate crystals are added until the upper surface of a crystalline layer is
on a level with the narrowest part of the tube in which the crystals are placed.
Cadmium sulphate solution is then added and the cells are exposed in a warm room
for a week or more before sealing. Some of the liquid evaporates, and many of the
fine crystals are loosely cemented together. This crystalline plug keeps the contents
in their proper places and enables the cell to be inverted.

Conclusions.

(1) Tli.- electromotive force of the Weston cadmium cell is the same whether it
contains electrolytic mercurous sulphate, chemically prepared sulphate, the salt as
precipitated by the dilution of hot strong sulphuric acid in which mercurous sulphate
is dissolved, or that resulting from the action of fuming sulphuric acid on mercury.

41(5 ME. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

(2) The size of the crystals of mercurous sulphate prepared by the first two and
the last of the above methods usually varies from 5 microns to 15 microns, and we
have found no electromotive difference between a saturated solution of these crystals
in a cadmium sulphate solution and a saturated solution of very large mercurous
sulphate crystals in the same solvent.

(3) The simplest method of preparing mercurous sulphate is by chemical precipita-
tion, and the product is more uniform than that of any other method so far examined.

(4) The electromotive force is the same whether the cadmium amalgam is prepared
by the electro-deposition of cadmium in mercury, or by the fusion of cadmium and
mercury. At normal working temperatures either a 10 or a 12£ per cent, amalgam
may be used.

(5) The electromotive force is probably constant over long periods of time, but the
cells should be compared with those of a standardising institution every 12 months ;
failing this, they should be compared with freshly set up cells.

(6) The change of E.M.F. with temperature may be calculated from JAEGER and
KAHLE'S equation or from that obtained at the National Physical Laboratory. As
the former has been used for so many years, we suggest its universal adoption.

(7) The small charging and discharging currents to which a cell is subjected in a
potentiometer circuit do not seriously affect the value of the electromotive force.

•
We desire to express our thanks to the Committee of the British Association for

grants of money for the purchase of materials ; to Dr. GLAZEBROOK for much advice
concerning the construction of the cells, and to Mr. J. A. SADD, of the Central
Technical College, and Mr. TINSLEY, for constructing standard cells to compare
with ours.

APPENDIX.

Added December 4, 1907.

ON THE COMPARISON OF THE ELECTROMOTIVE FORCES OF WESTON CADMIUM CELLS
PREPARED AT WASHINGTON, AT PARIS, AT BERLIN, AND AT TfiDDINGTON.

Dr. BURGESS of the National Bureau of Standards, Washington, journeyed to
Paris and Berlin after his visit to Teddington, and very kindly took with him a
number of cadmium cells from the National Physical Laboratory in addition to
others from Washington. Dr. F. A. WOLFF has forwarded us a report on the
measurements of the cells in Paris, from which we make the following abstracts.

MR. F. K. SMITH ON THE NORMAL WESTON CADMIUM CELL.

417

Eight of the American cells and eight of the English cells were compared at
the Lal)oratoire Central d'Electricitd under conditions which allowed of an approxi-
mation to 1 part in 100,000. The maximum deviation of the eight American cells
from their mean was found to be 0-00002 volt, and the difference between this mean
and the mean of the Weston cadmium cells of the Laboratoire Central was of the
order of 0 '00001 volt. The maximum deviation of the eight English cells was
about 0 00003 volt, and their mean E.M.F. differed from the E.M.F. of the French
cells by about O'OOOOl volt. A second set of comparisons, made at Paris in August,
confirmed the first measurements on the American cells.

Dr. BURGESS left four English cells at the Laboratoire Central, and M. JANET, the
Director of the Laboratory, has compared these with the French cells, with the
following results.

TABLE VII.

Approximate difference from mean in microvolts.

Cell No.

June, 1907.

July, 1907.

Octol>er,1907.

Novemlwr 20,
1907.

November 30,
1907.

P 52

+ 10

+ 1

+ 5

- 4

+ 1

P210

-10

- 4

-10

+ 1

+ 1

H 28

0

+ 1

-10

-19

-24

C 17 -10

+ 1

+ 15

+ 1

- 9

K 14

—

—

+ 11

+ 18

K 13

—

—

—

+ 6

+ 14

K 12

—

—

—

+ 6

+ 1

Mean E.M.F. of English

cells - mean E.M.F. of

French cells

+ 10

+ 24

+ 10

+ 14

+ 9

The cells K 12, K 13, K 14, were set up at the National Physical Laboratory on
October 2, 1907, and Mr. AOAR BAUOH kindly took them to M. JANET. The cells
H 26, H 28, and H 29 (see p. 402) were set up with pastes which may be very
slightly acid.

Eight of the American cells and eight of the English cells were compared at
Berlin, June 20 and June 21, while two American cells and four English cells remain
at the Reichsanstalt and have been intercompared from June 20 to September 30,
1907. The following statements are extracted from a formal report by Messrs.
JAEGER and LINDECK.

The results of the tests of the Weston cells brought over from America and
England are given in Table VIII. Since only the cells 183 and 184 from America,
and H 29, P 55, C 117, and C 12 from England, have been left in Charlottenburg,

VOL. ccvn. — A. 3 H

418

MR F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

the mean (indicated by A) of these six cells is taken as the basis of values given
in Table VIII.

TABLE VIII.

Western cells — America.

Weston cells — England.

Difference from A in microvolts.

Difference from A in microvolts.

Poll No

fall Tfn

June 20.

June 21.

June 20.

June 21.

19

- 3

- 7

H 26

- 3

- 7

105

-13

-18

C 19

+ 12

+ 5

P 8

- 8

- 8

P 53

+ 2

+ 1

P 9

- 8

-13

P 54

+ 2

+ 1

P10

- 8

-13

P 55

+ 2

+ 2

183

+ 7

- 3

C 12

+ 2

+ 4

184

- 3

- 3

C 117

+ 2

+ 7

187

- 3

- 3

H 29

- 8

- 6

Mean

- 5

- 8

Mean ....

+ 1

+ 1

It will be seen that the American and English cells agree very well among each
other. In the first series of measurements : Mean E.M.F. of English cells— Mean
E.M.F. of American cells =+0'000006 volt, and in the second series of measure-
ments the difference is +0'000009 volt. Two cells which Mr. KAYNER of the
National Physical Laboratory kindly took from Teddington to the Reichsanstalt, in
September, show equally good agreement.

The German cells mentioned in Table IX. (the series are designated by P, O,
and M) were prepared in March, 1907, with three different samples of mercurous
sulphate made by the von Steinwehr precipitation method. During the first few
months after their preparation these cells showed on the average a decrease in E.M.F.
of about 1 part in 10,000, and the E.M.F. had not attained a state of constancy at
the commencement of the measurements in question. Drs. JAEGER and LINDECK
suggest that the cause of this alteration lies in the fact that the three samples of
mercurous sulphate were washed out with dilute sulphuric acid in course of prepara-
tion. The relative agreement in the individual groups is, however, very good. In
the P group there are eight cells and the difference between any one cell and the
mean has at no time exceeded 2 parts in 100,000 ; in the O group there are six cells,
and in the M group six cells, and the corresponding differences for these groups are
about 1 and 17 parts in 100,000, respectively. Inter comparisons were made with
the older cells (1899) of the Reichsanstalt, which were taken as constant during the
period June 20 to September 30, 1907.

MR. F. E. SMITH ON THE NORMAL WESTON CADMIUM CELL.

TABLE IX.

Mean E.M.F. of the Single Groups — the mean E.M.F. of the American,
German (September 9, 1907), and English Cells.

419

Group.

Differences in microvolts.

June 20,
1907.

June 21,
1907.

July 11,
1907.

September 28,
1907.

September 30,
1907.

A. (2 American and 4 English)
B. (8 German P cells) . . .
C. (6 „ 0 „ > . . .
D. (6 „ M „ ) . . .

Mean German P, 0, M cells

-IB

+ 25
+ 54
+ 67

-10
+ 30

+ 57
+ 71

-13

+ 20
+ 48
+ 54

-21
-12

-12
- 6

+ 21
+ 14

+ 49

+ 53

+ 41

—

+ 10

As will be seen from the above table, the cells from America and England have
remained constant during the period June 20 to September 30 ; the German cells
of Groups P, O, M, have, however, decreased 4 parts in 100,000. Drs. JAEGER and
LINDECK think that it is not improbable that the alteration will continue, but owing
to the slightness of the change this can only be tested after long periods.

It would seem, as the result of the last measurements on September 30, that
the differences between the various cells compared were, at that time, only a few
parts in 100,000. By making use of the average value of the cells P, O, M, obtained
at this time, and taking into consideration the data given in Dr. WOLFF'S report
(and part of that given on p. 403 of this communication), Drs. JAEGER and LINDECK
give the following differences for the cells of the different countries, the figures being
rounded off to the hundred-thousandth part :—

E.M.F. of English cells

(Mean of more than 100 cells)

E.M.F. of German cells
(Mean of P, O, M cells)

E.M.F. of German cells

}-
}-

E.M.F. of American cells 1
(Mean of 12 cells)

E. M. F. of English cells =±0x10

-* volt,

~8

- E.M.F. of American cells = + 1 x 10~*

As the French cells are also in good agreement with the American and English
cells, considerable advance would appear to have been made with the standard cell
question.

The English cells, H 26, C 19, P 53, and P 54, were received at Washington on
August 22. On the same date C 19 was about 40 microvolts higher than its
companion cells, but on August 27 a comparison led to the results given in

3 H 2

420

MR. F. E. SMITH ON THE NORMAL WKSTOX CADMIUM CELL.

Table X. Dr. WOLFF has also forwarded the results of comparisons made by him
on P 8, P 9, P 10, and 187 (American), of PCN 4 and PCN 6 (French), and O 1 and
O 2 (German). Unfortunately, the depolariser in the German cells was disturbed in
transit, and the results obtained are not, therefore, given in Table X.

TABLE X.

August 27, 1907. E.M.F. of Cell-Mean E.M.F. of the American, French,

and English Cells.

American

P8 - 2xlO-6 volt

P9 - 5xlO-«

P10 - 5xlO-«

187 + 5x10-°

+ 16xlO-«
PCN 6 - 18 x 10-«

H 26 - 5 x 10-«

C 19 + 5 x 10-«

P53 + 4xlO-6

P 54 + 5 x 10-"

Mean = - 2 x 10~6 volt

Mean= - 1 x lO'6 „

Mean = + 2 x 10'° „

We heartily thank the various gentlemen who have assisted in these comparisons.

[ 421" J

XI. Electric Furnace Reactions under High Gaseous Pressures.

By R. S. HUTTON and J. E. PETAVEL.
Communicated by Profesxor A. SCHUSTER, F.R.S.

Received January 31, — Read March 7, 1907.
[PLATE 6.]

CONTENTS.

r n

Introduction 421

Description of apparatus . . . . 422

Large high-pressure furnace . . . » 422

Carbon feeding mechanism . . . . 424

Carbon holders 426

Windows ....... ... 426

Valves and gas connections ... 427

Small furnace for high-tension currents 428

Ga« preparation and compression 431

General observations on the electric arc under high gaseous pressures . . 431

On the formation of calcium carbide 437

On the fusion of silica 444

On the formation of carborundum .... 445

On the direct reduction of alumina by carbon ... 44C

Tables I. to X .... 451-462

INTRODUCTION.

SOME ten years ago the classical work of HENRI MOISSAN laid the foundations of the
scientific study of high-temperatmre chemical reactions.

It is hardly necessary to recall the rapid and extensive development which the
subject has since experienced ; it must, however, be remembered that the progress
has been almost exclusively along technical lines, and even at the present time very
little detailed work on the chemical "and physical sides of the question has been
published.

Doubtless individual inventors have acquired extensive experience and knowledge
each of his special branch of the subject, but they have seldom found it advisable to
impart the results of their researches.

The field of investigation, even with regard to the purely chemical phenomena

VOL. ccvii.— A 423. 17.1.08

422 MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

occurring in quite ordinary cases, is far from exhausted, and practically nothing is
known of the modifications introduced by abnormal conditions, such for instance as
the combination of high pressure and high temperature.

The present research has been undertaken with a view to determine the correlation
between the physical conditions and the chemical effects in the electric furnace, and
more especially to ascertain the direct results produced by high gaseous pressures.
In such work, if it is to be of real value, it is necessary to carry out the experiments
on as large a scale as the limitations of cost and labour will permit ; it is also
particularly desirable to provide means for the modification of the various factors
within the widest possible range. The chief factors being power, current, electro-
motive force, and above all, pressure.

In work carried out during preceding years under atmospheric pressure the
necessity for employing now one type, now another of electric furnace determined us
to select an apparatus suitable for both open and smothered arc, as also for resistance
heating.*

The above considerations accentuated the already somewhat difficult task of
designing a furnace suitable for high gaseous pressures. For, as already suggested,
it would have been of little use to provide for a slight increase of pressure over that
of the atmosphere.

The apparatus constructed is capable of employment for electric heating according
to the most varied type of furnace, and has frequently been used for pressures as high
as 200 atmospheres. We were thus in a position to extend our direct experimental
study up to the limits which engineering difficulties set to practical application.

With regard to the experimental work, the first step was to investigate the
additional effect of high pressure upon the more characteristic electric furnace reactions,
it being obviously advisable to start by repeating the better known preparations,
retaining as far as possible all other conditions similar to those at present in use.

It is with a general investigation of this character that our communication has
to deal.

To avoid burdening the description of the work with the many numerical results,
we have collected these in tables in an appendix to the paper.

DESCRIPTION OF APPARATUS.
Large High-Pressure Furnace.

From what has been said above it will be clear that an apparatus was necessary
capable of being adapted to very varied requirements.

This end was met by designing a large steel enclosure of about 20 litres capacity,

* We have frequently had occasion to divide the enclosure into a number of separate chambers, e.g., for
absorbing gaseous products inside the furnace, as also for a condensing chamber in volatilisations ;ind
distillations.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

423

provided with various fittings for the introduction or circulation of gas, gauges for
measurement of pressure, windows for olxservation and, finally, with insulated carbon
holders leading the current to the inside of the furnace. Within this shell many
different forms of furnace, such as vertical or horizontal arc or resistance core, could
be built up as illustrated in fig. 1 1 (p. 438).

The construction of the enclosure will be easily understood by reference to tigs. 1
and 2, giving sectional drawings. The shape of the interior is cylindrical, 10 inches

Fig. 1. Sectional drawing of large pressure furnace.

A. Main forging, 1£ inches ruling thickness increased to 2 inches over central l»elt, through

which the various openings are bored.
C. Water jacket surrounding the body of the furnace.
H. Water jacket surmounting the cover.

B. Cover held down by ten 2J-iuch studs, the joint being made by a lead ring placed in

the spigot groove S. The projection N protects the joint from contact with the hot
gases when the furnace is in use and shields it from mechanical injury while the cover is
lii-ing lifted or replaced.
L. Cast-iron lining.

diameter by 17 inches long, with hemispherical ends, one of which forms the cover B
and is held in place by ten 2^-inch studs (F,, F2) which are fixed into a flange of the
main forging. The cover is rendered gas-tight by a spigot joint S, packed with lead ;
it is surmounted by a cast-iron casing H, through which cooling water was circulated.

The main forging A is surrounded by the cast-iron water jacket C.

Both the hemispherical ends of the furnace have projections K,, Ka bored out to
a distance of 3 inches.

•

The carbon holders which move in these recesses are thus protected from the direct
heat or flame of the furnace. The length of the projections KI, K, is sufficient to allow

424

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

a feed of 8 inches. To obviate any risk of damage to the main forging by contact
with the hot furnace materials, a cast-iron lining L was always used.

When in a horizontal position the furnace rests on four cast-iron feet (not shown in
the figure). When vertical it is supported by the cover of the water jacket, the
lower carbon feeder passing through a hole cut in a massive wooden stand.

Fig. 2.

Sc*ic IN INCHES

Transverse section through the centre of the large furnace.

W. Water jacket.

F. Main forging.

L. Cast-iron lining.

The inlet valve is screwed into A, whereas the openings B and C receive the windows shown in
fig. 4, outlet valves and gauge connections, or, when required, auxiliary insulated
terminals.

The main forging is provided with three openings, as shown in fig. 2, which is
a section through the centre of the furnace perpendicular to the axis of the carbons.

The aperture A served to receive the valve through which the enclosure was filled
with compressed gas, whereas in most cases one of the windows shown in fig. 4 was
screwed into B. The third opening was connected to a pressure gauge and served
also, when desired, for the escape of the gaseous products of reaction.

Carbon Feeding Mechanism.

This is shown in detail in fig. 3. A ring B is fitted to each of the projeptions K of
the furnace. To this ring the small cover A is bolted. The joint is made, as in the
case of the main furnace cover, by means of the lead-packed spigot V. The cover

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

carries two columns Ci, Ca surmounted by a yoke Y which is insulated from them by
micanite bushes and washers II. To this yoke 'the main terminals (not shown in the
figure) are fixed. The nut N is revolved by means of steel levers inserted in the
holes T, and thus the feeding rod S is moved forward. The feather F fits in a groove
cut in this rod and prevents it following the rotary motion of the nut.

Fig. 3. Carbon feeding mechanism of the large furnace.

The email cover A is bolted down to the ring B, which is fitted on to the projection K, forming

part of the main forging.
Two columns, Ci, C->, support the insulated yoke Y, in which the nut N is revolved by steel

levers fitting into TI and T2.
A stream of water flows through the hollow feeding rod S, the nipples I and 0 carrying the

inlet and outlet pipes. A central pipe can be used to pass a flow of compressed gas

through the axis of the electrode M.

The glands Q of the stuffing box P are electrically insulated from the cover A.
The electrode M is soldered into the cup-shaped holder H which screws on to the end of the

feeding rod.

The feeding rod passes into the furnace through the insulated stuffing box P.

This stuffing box serves the double purpose of making a gas-tight joint and
providing insulation sufficiently perfect for the relatively low electromotive force
which is generally required with this furnace. As packing, a mixture of asbestos and
tallow is used, which in itself assists the insulation.

The stuffing box is compressed by means of the ring D, which presses on the gland
Q, but is electrically insulated from it by mica washers and bushes. The inner gland is
insulated iu a similar manner from the steel cover.

The feeding rod is hollow. The current of water passes into it at I, and flowing
through an inner brass tube is delivered at the extremity of the rod and passes back
to the outlet O. A gas connection G is also provided by means of which compressed
gas can, when necessary, be passed directly into the centre of the furnace through the
axis of a hollow electrode (M).

VOL. ccvii. — A. 3 I

426

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

Carbon Holders.

These are of two different patterns, according to the size of the electrode used.

Carbons of 30 millims. diameter or less are held by clamps similar to those used
with the smaller furnace, as shown at T in fig. 6 (p. 429).

For larger carbons the limited space available prohibits the use of such an arrange-
ment, which in any case would hardly provide a sufficiently good contact for heavy
currents (500 to 1000 amperes).

The larger carbons are therefore electro-coppered at their ends, and soldered into
cup-shaped holders, as shown at H (fig. 3). The lip of the cup is fitted with a ring
of refractory insulating material, U, which nearly fits the bore of the tube K, and
thus protects the stuffing box from flame and dust.

Fig. 4. Windows.

A and B. Gun-metal fittings carrying the glass windows W.
F. Steel wall of furnace enclosure,
a. Gas-tight ring joint.

R. Ring making water-tight joint between the fitting and the water jacket K.
The design A is used when working with arcs of small intensity, and B for larger currents.

Windows.

Although unnecessary for the purely chemical work, it was of considerable
importance to be able to observe, project, or photograph the arc itself or its spectrum
under conditions of high pressure.* By providing two openings (B and C, fig. 2)
diametrically opposite, absorption spectra could be observed during the operation of
the arc.

The forms of construction are shown in fig. 4. The window itself consists of a glass
or quartz cone (W) f inch thick and £ inch diameter at its smaller end. This

* " Preliminary Note on the Effect of Pressure upon Arc Spectra," J. E. PETAVEL and R. S. HUTTON,
1 Phil. Mag.,' Nov., 1903, vol. 6, pp. 569-577.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES. 427

transparent cone is forced into the gun-metal fitting after being surrounded with a
tli in film of cement, and is held in place hy a metal ring, the shape of the glass
tending to make the joint more perfect the higher the pressure.

The joint betw een the fitting and the aperture in the main forging is made hy the
ring a turned on the end of the fitting, which presses tightly against a steel ledge, no
packing being required.

The joint between the fitting and the water jacket is made by means of a gun-
metal ring R, which screws on to the fitting itself.

The two types of window differ only in the relative position of the transparent
plug. The design shown at A gives a clearer view of the arc, but for very large
currents it is advisable to make use of the fitting B, in which the glass plug is more
carefully protected from the source of heat.

Valves and Gas Connections.

The types which have already been described* were employed for regulating the
flow of the various gases used.

After the construction of the furnace was completed it was tested to 450 atmos-
pheres, and has since been used frequently up to 200 atmospheres gaseous pressure.

At first it was anticipated that so large a joint as that of the main cover would
show some leakage at the higher pressures, the total stress on the bolts retaining the
cover amounting under ordinary working conditions to over 100 tons. These fears
were, however, not realised, the only precautions necessary being to keep the joint
perfectly clean and, of course, to tighten up the nuts evenly all round. Occasionally,
as the pressure rose, a slight escape of gas was noticed, but this was stopped without
any difficulty by tightening up the corresponding stuffing box or joint.

In fact, it may be said that throughout the work no difficulty has been experienced
in keeping the apparatus gas-tight.

The furnace is, of course, equally suitable for work in vacuo, and has occasionally
been used in this way — in connection, for instance, with spectroscopical investigations.

In the course of the present research the electrical conditions have varied widely,
in some cases as much as 1000 amperes, in others 500 volts, having been employed
without difficulty. The power used in most experiments was between 10 and
15 kilowatts. When it is desired to use high-tension currents (1000-25,000 volts)
with this furnace the carbon feeding mechanism of the small apparatus can be used
to replace that described above, with which it is interchangeable.

The weight of the enclosure was, of course, considerable, and to facilitate its
manipulation a crane was fixed to the main laboratory wall The crane is regularly
employed for the removal of the cover, and serves also to lift the entire furnace and
change its position from horizontal to vertical, or vice versd.

* J. E. PETAVKL, ' Phil. Trans.,' A, vol. 205, p. 369, 1905.
3 I 2

428

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

Small Furnace for High -Tension Currents.

Since there are a considerable number of electric furnace reactions, as, for instance,
most gas reactions, for which very high-tension currents are necessary, it was decided
to complete the equipment by providing a furnace specially for this class of work.

In this case a large capacity, instead of being an advantage, would constitute a
serious drawback, owing to the difficulty of preparing and purifying such a quantity
of gas. Moreover, in the case of a high-tension arc, the natural temperature gradient
is so steep that the maximum temperature is easily and safely attained even in a
small enclosure. The rapid rate of cooling which results from the proximity of the
arc to the cold walls of the furnace is also of material advantage.

Fig. 5. Sectional view of small furnace.

The body of the furnace is suspended from a cast-iron plate A bolted to a wooden stand B.

The two ends KI, K2 are closed by covers which carry the feeding mechanism (see fig. 6). The body

of the furnace is surrounded by a water jacket W.
A little below the centre the walls are thickened up to 2 inches and are pierced by two openings.

The inlet valve screws into H, and a window (see fig. 4) into G.

The smaller furnace is of one-tenth the capacity of the larger, and consists of a mild
steel cylinder about 3 inches internal diameter, with walls \\ inches thick, surrounded
on the outside with a water jacket (see fig. 5). The walls are thickened up a little
below the centre, and two openings are bored in the ring thus formed. One of these

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

429

(H) serving for the inlet valve, the other (G) for receiving one of the windows we
have already described, the size and form of the openings being such as to render
all fittings interchangeable with those used in the larger furnace.

SOLI IN IXCMII

Fig. 6. Carbon feeding mechanism for the small electric furnace.

The ring B screws on to the end K of the furnace and supports the cover A. Into this are

screwed the columns C which carry a thick plate of insulating material M.
The feeding screw Q is rotated by means of a con! passed over the pulley W ; this screw works

in the nut P which is fixed in the centre of the insulating plate (M).
The electrode X is held in a clamp T which is fixed to the feeding rod R.
This rod passes into the furnace through the stuffing box S, which forms the upper part of the

central steel plug E. This plug is forced by the nut N against a cupped insulating piece

D which fits a recess in the furnace cover.

430

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

The chief characteristic of this furnace lies in the means employed for insulating
and moving the carbon holders. The mechanism is clearly shown in figs. 6 and 7.
A conical hole tapering outwards is bored in the centre of the covers of the cylinder,
and receives a cup-shaped insulating bush (D) of ebonite or red fibre. & central
plug E is provided with a mushroom-shaped end which fits closely into this cup and

Fig. 7. Small high-pressure furnace.

is lightly drawn against it by means of the nut N placed on the outside, this nut in
turn pressing on the insulating ebonite washer I.

The gas pressure itself forces down this plug firmly on its seat and secures a
satisfactory joint. The upper end of the steel plug contains the necessary stuffing
box (S) through which passes the feeding rod R.

From each cover of the furnace project three steel pillars, carrying at their
extremities a thick triangular plate M of insulating material. To the centre of this

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES. 431

IB fixed the nut (P) in which the screw-gear works. This screw-gear is rotated by
means of a cord passing over the pulley W. Thus, when the furnace is being used
for very high electric tensions, the feeding gear can be manipulated from a safe
distance. This form of insulation has been found to work satisfactorily up to
3000 volts. For higher electromotive forces one of the steel feeding rods is removed
and replaced by a thick-walled glass tube. The current, which under these conditions
is of course very small, is led by a number of fine platinum wires fused through the
glass to the electrode, which is supported on the end of this rod.

With this modification the furnace has been used up to 25,000 volts.

Gas Preparation and Compression.

In connection with this research it was necessary to manufacture and compress
relatively large quantities of hydrogen, carbon monoxide, ethylene, and nitrogen.

The question has been dealt with fully in a recent publication to which we must
refer those specially interested in this branch of the subject.*

GENERAL OBSERVATIONS ON THE ELECTRIC ARC UNDER HIGH GASEOUS

PRESSURES.

At the time the present research was started very little information was available
with regard to the electric arc burnt in a compressed atmosphere, the investigations
having been limited to small arcs and pressures of about 15 atmospheres. The belief
was then current that it would prove to be impossible to maintain an arc under
gaseous pressures of 100 or 200 atmospheres.

This erroneous conclusion was due to a misconception of the nature of the
difficulties which had been encountered by the various workers, t

For spectroscopical investigations it is necessary to obtain a relatively long arc
giving a clear flame between the extremities of the electrodes. Such an arc, it is
undoubtedly true, can only be maintained in a dense atmosphere by means of an
exceptionally high electromotive force.

It should nevertheless be clearly understood that even low-tension arcs will burn
without difficulty. At ordinary current densities the entire phenomenon is, however,
confined to the crater itself, and a projected image of the arc shows merely the
outline of the white-hot crater, and only occasionally is a flash of flame visible on the
periphery.

Electrically the arc is still well characterised by its high electromotive force, which
instantly disappears if the electrodes are brought into actual contact.

* BUTTON and PETAVKL, ' Journ. Soc. Chem. Ind.,' 1904, vol. 23, pp. 87 to 93.

t W. E. WILSON and G. F. FIT/GERALD, ' Roy. Soc. Proc.' 1896, vol. 60, pp. 377-383.

432

MESSRS. R. S. BUTTON AND J. E. PETAVEL ON ELECTRIC

When the arc is started in a compressed oxidising atmosphere, the current is at
first unsteady and the electrodes must be rapidly fed up. Soon, however, a steadier
state is reached. If, after such a run, the furnace is opened and the carbons
examined, it will be observed that, by the action of the current, the electrodes

have been so shaped as to nest one into the other as shown
by the dotted lines in fig. 8. In this way a considerable
cross-sectional area is produced over which the discharge can
occur.

The E.M.F. of the arc rises as the pressure of the surround-
ing atmosphere increases* and at the high pressures used in
the course of this work it becomes more than double the
normal value.

It is, however, the first few atmospheres which produce the
greatest effect upon the voltage.

A detailed analytical investigation of the arc would in
itself, as can be judged from the great amount of work
carried out at ordinary pressures, require considerable ex-
penditure of time and is outside the scope of the present
research. We therefore limit ourselves here to the few
observations recorded in Table I. (p. 451), in which the
behaviour of the arc under certain definite conditions of
pressure and current is recorded.

Fig. 8. Configuration of car- An interesting effect is throughout noticeable. Although
bon electrodes after use ,L i ,, /. , -,

the maximum length or arc is so much reduced, the voltasre is
m an inert gas under ' ,11

in aJl cases abnormally high, and consequently a large amount

of power is concentrated in a small space.

Two very distinct types of arc exist. The most usual in
to the shape of the crater these enclosed furnaces is found with a non-oxidising atmos-
of the positive and pro- phere such as carbon monoxide or nitrogen,
jects into it as shown by jn 8UCh arcg at ordinary current densities the electrical

conditions are complicated by the rapid growth of a deposit
A mushroom-shaped deposit .

of carbon accumulates on °f Carbon' chiefly around the negative electrode and often
the extremity of the nega- completely enclosing the end of the positive electrode. The
tive electrode. difference between a resistance and an arc is then less marked.

The arc flame is not visible, but is replaced by a zone of

brightly incandescent carbon ; electrically the conditions are ill-defined and difficult
to reproduce. It is consequently only during the early part of the experiment that
concordant measurements can be obtained. As time goes on the carbon deposit
builds itself up, the voltage shows a tendency to rise, and the general appearance

* DUNCAN, ROWLAND, and TODD, ' Electrical World,' 1893, vol. 22, p. 101, for 6-ampere arcs up to
10 atmospheres pressure.

high pressure.

The end of the negative
carbon becomes moulded

I I RNACE REACTIONS UNDER HIGH OASKors I'l.'KSSURES.

indicates that the electrical conditions more nearly approach those of a resistance
than of a true arc.

It is, however, almost inconceivable that a power of 5 or 10 kilowatts (see fig. 10,
B and C) can be expended in such a limited volume of solid material without
volatilising it; and, as we shall see, the amount of disruption as evidenced by the
feed required is under these conditions extremely small.

Negative
electrodes.

Positive

divtrodes.

123 4

Fig. 9. Photograph of carbon electrodes after use. (Horizontal arc.)

No.

1
2
3
4

Pressure, Current,
atmospheres, amperes.

30
30
50
48

60
400
100
120

E.M.F.,
volts.

60

65

100

130

Gas.

Originally air, but all oxygen fixed by carbon.

Ditto.

Coal gas.

Originally air, but all oxygen fixed by carbon.

With an exceptionally high current density the arc in a non-oxidising gas at high
pressure gives a well-defined flame.

In fig. 10, D, a comparison is given between arcs in carbon monoxide at 11 and 16
atmospheres and an experiment made by Mrs. AYRTON* with an enclosed arc at
ordinary pressure, using carbons of similar size. It will be seen that the increase of
voltage due to an increase of pressure of 1 o atmospheres is very considerable. For

VOL. ccvii. — A.

* Mrs. AYKTON, ' The Electric Arc,' p. 304.
3 K

434

MESSRS. R. S. HUTTON AND J. E. PKTAVEL ON ELECTRIC

these high current densities the apparent resistance of the arc under pressure remains
positive just as at atmospheric pressure.

An entirely different type is obtained in an oxidising atmosphere, and in this case
alone are the results comparable with the well-known conditions of the ordinary
open arc.

In fig. 10, E, the E.M.F. of a 150-atmosphere arc of about 2 millims. length is given
and compared with measurements made at atmospheric pressure, the positive carbon
being 41 millims., the negative 27 millims. diameter.

80

*

lu

160
140
120
100
80
60
40
20

120

too

80
60
10
20

A ARC IN AIR
A

* ~ 9

AT ATMOSPHERIC PRESSURE.

5

100 ISO 200 250
i • i , i i i i i

CURRCHT //v 50 AMP fata

\ i i i i i

B

C ARC IN NITROGEN

^-^^ ° ATMOSPHERES

-120
-100
-80
-60
-40

~fflIO 20 3

yssz"

ARC IN
CARBON MONOXIDE

0 40 50 60 70 80 90

I i I i 1 1

50

1 1 1 1 ! 1

100 ISO

D ARC IN CARBON MONOXIDL

£ ARC IN AIR

•-'Vj

£~LECTffOMOT/l/£
FORCE

/wo
CuffffEMT OF THE
/IffC //V l//!ff/#(/S

GftSES.

1 1 IM* CtRBOits > ^^

*^St--»

e

-100 X-Sfa
-60s '^

" 10 20-30 40 50 60 70

HU g -» Ir

' AMffHtf-* 50 100

i . i i i i i i i i

Fig. 10. Curves of the electromotive force and current of the arc in various compressed gases.

The data refer to the conditions existing while a fair proportion of the oxygen was
still present ; the arc then shows a bright flame which, if the electrodes are fed up
rapidly and with regularity, can be easily maintained. Here again the increased
voltage observed is due principally, as shown by other experiments, to the first 10 or
15 atmospheres.

For the purpose of general comparison, few data on large current arcs being
available, a number of measurements were made with an open arc burning at
atmospheric pressure between carbons (positive 41 millims., negative 27 millims.) and
maintained at a constant length of 8 millims. The curve thus obtained is recorded in
fig. 10, A.

Finally it should be mentioned that, in the course of the chemical work to be
described, constant use has been made of " smothered " arcs, as, for instance, in the

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURlH 435

production of calcium carbide. With arcs of as much as 500 amperes between
the extreme limits of our experiments, i.e., from 0'5 atmosphere up to nearly
200 atmospheres, no consistent effect on the voltage which could be ascribed to
the influence of gas pressure was apparent.

A noticeable feature in all the experiments carried out in air is the very rapid rate
of burning of the electrodes.

To take one instance : with 27-millim. carbons and a 30-ampere arc under a pressure
of 29 atmospheres the carbon was consumed at a rate of about 6 millims. per minute,
which is nearly twenty times as fast as at ordinary pressures.*

This burning is principally limited to the positive electrode, which in the above
experiment was consumed seven times as fast as the negative, the relative rates at
ordinary pressures being about as 3 : 1. At a still lower current density the negative
carbon shows practically no loss of weight.

Under the normal conditions of most of our experiments, that is with very high
current densities, the combustion becomes very violent.

The oxygen thus becomes rapidly exhausted and the subsequent feeding required
is relatively very small.

At first carbonic acid is formed which is in turn reduced to carbon monoxide. This
reduction occurs also when the arc is run in an atmosphere ot compressed carbonic
acid. In the latter case an interesting observation was the separation of flocculent
carbon which was seen to be moving rapidly in the convection currents. This
phenomenon is observed from the commencement of the experiment.

The decomposition of the carbonic acid under these conditions is comparatively
slow, in one experiment less than half being decomposed by the end of half an hour.

A somewhat similar process goes on also in non-oxidisiiig gases. Finely divided
carbon is deposited in considerable quantities on the cool walls of the enclosure, the
weight collected being in fair agreement with the loss from the electrodes. With
such gases the atmosphere, however, remained perfectly clear, t

* B. MONASCH, ' Der elektrische Lichtbogen,' p. 22.

t In cases in which a hydrocarbon gas atmosphere («.</., coal gas) is employed, the deposition of carbon
is augmented by the breaking up of the higher hydrocarbons.

The following analyses give the constitution of the coal gas before and after a run at about
60 Atmospheres, which lasted 36 minutes with a power of about 10 kilowatts, a horizontal arc being
employed. In this case both electrodes had increased in weight, the positive by 6, the negative by

16 grammes : —

Before run. After run.

Hydrocarbon vapours 1 • 0 0-0

Heavy hydrocarbons 4 • 2 0 • 7

CO, 2-0 0-6

CO 15-0 14-7

CH< 18-0 -i '.*

H 42-0 41-8

0 1-7 0-0

N (by difference) ... 16- 1 17-3

100-0 100-0

3 K 2

436 MESSRS. R. S. HUTTON AND .1. E. PKTAVKI, ON KI.KCTKTC

The interesting results thus obtained with regard to comhustion led us to carry out-
some experiments on the oxidation of electrodes of other materials, the production of
an atmosphere free from oxygen and the oxides of carbon being also of considerable
practical importance for the further work we had in view. A priori one would be led
to believe that copper, iron, or aluminium when heated and fused in highly compressed
air (100 atmospheres) would not only rapidly fix the available oxygen, but would do
so with sufficient intensity to make the combustion self-supporting.

Repeated attempts were made to produce this result, a summary of which will be
found in Table II. (p. 452).

Briefly speaking, we may say that with an iron bar maintained for one hour at a
bright red heat and then partially melted by means of a current rising to 1000 amperes,
the percentage of oxygen fixed was hardly appreciable.

A similar result was obtained in the case of copper, whereas even aluminium melted
in the arc at a pressure of 26 atmospheres only oxidised on the surface, the well-known
tenacity of the oxide films being sufficient to prevent the rapid combustion of the
metal.

The negative results thus obtained led to some experiments being carried out with
oxygen.

An arc was struck between two iron bars of 1^-inch diameter, surrounded by
oxygen at 1 5 atmospheres pressure ; as soon as the temperature reached a bright red
heat, a vivid combustion commenced on the positive electrode and continued quite
steadily, although the current was then cut off.* After the combustion had proceeded
a short time the pressure was gradually reduced, the combustion ceasing when the
pressure had fallen to about 5 atmospheres. A length of some 5 inches of the bar had
by this time been consumed, the loss of weight being 535 grammes, the loss on the
negative electrode being only 15 grammes. The product of the reaction was collected
in a crucible placed for this purpose under the arc, and on analysis proved to be
magnetic oxide (Fe3O4).

It is worthy of note that although the partial pressure ot the oxygen in the
experiments previously referred to was considerably above that required in the case
of the pure gas, the combustion was not merely insufficient to maintain the temperature
required for continued combustion, but even with the assistance of the arc the total
iron burnt was almost negligible. The result is due probably to the high effective
heat conductivity which is characteristic of compressed gases, the evolution of heat
due to oxidation being, in the diluted gas, insufficient to overcome this cooling
effect.!

It was thought also that oxygen might easily be removed from the atmosphere by
the introduction of successive small quantities of hydrogen, care being taken to always

* E. FRANKLAND, ' Journ. Chera. Soc.,' 1864, vol. 17, pp. 52-55, describes an interesting case of the
combustion of iron, in compressed oxygen.

t J. E. PETAVKI, 'Phil. Trans.,' A, vol. 197, pp. 229-L'54, 1901.

FURNACE REACTIONS UNDKR Hlcil <:. \sEOUS PRESSURES. 437

keep ln-|ci\v the explosive limit, a small air. l>eing maintained to effect the combination.
Under these conditions, however, the water vapour produced was rapidly converted
into carlxHi monoxide and hydrogen under the action of the arc. The method had,
therefore, no advantage over the direct combustion by carbon, and was al>andoned.

In many of the cases in which a carbon arc was maintained after all the oxygen of
the air had been fixed, small traces of hydrocyanic acid were detected. In the above
experiment, in which hydrogen was present in considerable amount, the formation was
much increased.*

|

ON THK FORMATION OF CALCIUM CARBIDE.

The production of calcium carbide, constituting a simple and typical example of
electric furnace reactions, was considered a suitable subject for the first series of
experiments.

From previous experience we were impressed with the necessity of maintaining the
greatest possible uniformity in the conditions under which the furnace was operated.

Above all it was desirable to avoid the variations introduced by the use of different
forms of furnace construction, and therefore for all the experiments dealt with in this
section we have employed the simple type represented in fig. 11, A, the dimensions
of the furnace, the size of the electrodes, the weight and constitution of the reacting
mixture l>eing kept the same and, as far as possible, the factors under consideration
varied only one at a time.

The building up of the furnace entailed the use of the enclosure iu a vertical
position as shown in Plate 6, fig. 1. The furnace cover having been removed, by
means of a crane installed for the purpose, the cast-iron liner is raised and deposited
on a separate stand where it is prepared for the experiment. To protect the bottom
of this receptacle from the direct action of the arc, a layer of powdered retort carlxm
is first introduced, which thus constitutes the lower electrode. The mixture of lime
and carbon (about 10 kilogs.) is then filled up around a paper tube which serves to
keep a central passage free for the upper electrode.

If there is any doubt as to the perfect desiccation of the raw materials, the cast-
iron pot with its contents must be maintained at a red heat for some hours before it
is placed in the enclosure. This not only ensures more consistent results, but renders
it possible to follow the progress of the reaction by a measurement and examination
of the gases generated. The cover of the enclosure is lowered carefully into position,
the carbon electrode sliding into the cylindrical space which has been reserved for it.

After the bolts have been tightened up and the desired quantity of gas introduced,
the arc is started by lowering the upper electrode, which then comes in contact with
the carbon lied beneath it.

* H. HOYKRMANN, ' Chcni. Zeitung,' 190:.', vol. 28, pp.70, 71 : .1. URTS/KIKWIC, '%. fur Klektrochwnie,'
1903, vol. 9, pp. 83-«5 ; H. AuKR, ' Acad. Sci. Buda-Pesth,' 1904.

438

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

During the experiment the voltage and current are read at frequent intervals ; the
watts, at these instants, plotted on a curve enable the total kilowatt hours supplied
to the furnace to be obtained by integration.

B

A.

B.

D

c

-b

-a

Fig. 11. Sectional views of various types of electric furnace.

A. Smothered arc (before run).

I is the cast-iron liner in which the charge was placed, h, cover of same, e, vertical
carbon electrode (41 millims. diameter), d, granular carbon bed forming the lower
electrode, c, charge.

AI. Smothered arc (after run),

a, ingot of fused product, b, fused and fritted material forming walls of cavity, c, unacted-
i m material.

B. Itesislance (before run).

e, carbon electrode. /, graphite end-piece leading current to core, g, resistance core of
granular material or carbon rod, or other solid " resistor." c, charge, d, granular
carbon bed or other form of lower electrode.

BI. Resistance (after run).

a, ingot of fused product, b, fused and fritted material forming walls of cavity, c, unacted-
on material.

C. Horizontal arc : radiation heating.

«i, e2, electrodes. /, walls or jacket of heat insulating material, c, charge in carbon or
other crucible.

D. Smothered arc unth two carbons embedded in the material (used horizontal or vertical position).

e\, «a, electrodes, r, charge.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSUBiB. 439

As the reaction proceeds, the pressure in the furnace, due to the evolution of carbon
monoxide, rises rapidly and the fused carbide collects upon the bed of carbon and
soon accumulates sufficiently to come in contact with the end of the electrode. The
sharp drop of the electromotive force thus produced serves to indicate that the time
has arrived to raise the carbon slightly.

There is no difficulty in maintaining these smothered arcs even at the highest
pressures, and the regulation of the power is quite a simple matter. Currents of some
500 amperes have frequently been employed in the course of this work and maintained
as long as desired.

It is very noticeable that, contrary to experience with such arcs when maintained
at atmospheric pressure, the upward rush of the gases through the finely divided
charge causes very little* displacement of material. For a given rate of reaction the
velocity of the gas currents is of course roughly in inverse proportion to the working
pressure. At high pressures, therefore, the gaseous products of reaction rise at
a relatively slow speed and percolate through the mixture without disturbing it.
When, on the other hand, the enclosure is evacuated the projection of the material is
very marked and causes considerable trouble.

An extremely low consumption of the carbon electrodes is characteristic of these
enclosed furnaces, the deterioration of the electrodes being so slight that they can be
repeatedly used In general practice the loss in weight of the electrodes is an
important question and in favourable cases is still between 1 and 3 per cent, of the
output of carbide. The consumption in ordinary furnaces on a laboratory scale is still
larger, whereas in the present experiments the loss has always been so small as to be
hardly appreciable.

We may therefore conclude that the corrosion is not due to the dissociation of the
lime as suggested by GIN,* but is to be ascribed to atmospheric oxidation.

A summary of the more important experiments will be found in Table III.,
whereas Table IV. gives the detailed observations referring to one typical can.

Before discussing these results it may be well to consider briefly the mechanism of
the chemical reaction upon which the formation of the carbide depends.

It is generally stated that carbon first commences to react readily with lime when
the latter reaches its melting point, the production of carbide below this temperature
being limited and of little practical importance.! '• "

In a careful investigation of ROTHMUNDJ it has however been shown that a definite
equilibrium exists at about 1600° C., as represented by the equation

CaO + 3C^±

the reaction tending to go from right to left at higher, from left to right at lower

* G. GIN, ' Z. fiir Elektrochemie,' 1902, vol. 8, p. 397.

t H. MOISSAN, 'Comptes Rendus,' 1904, vol. 138, pp. 243-245.

| V. ROTHMUND, ' Z. fur anorg. Chemie,' 1902, vol. 31, p. 136.

440 MESSRS. R. S. BUTTON AND .T. E. PETAVEL OX ELECTT?TC

temperatures than this. The formation of carl>on under these conditions has
been observed by A. FRANK.*

From this it. would at first sight appear that, if the carbon monoxide resulting
from the formation of calcium carbide were retained in the furnace and the pressure
allowed to accumulate, the reaction would soon come to a standstill.

Our experiments are, however, in direct opposition to this conclusion, proving that
the temperature prevailing in the furnace is sufficiently far above the point of
equilibrium to preclude the inverse reaction so long as the heating is continued.

On the other hand, the above considerations alone might lead one to suppose that,
already at any temperature above 1GOO°C., the formation of carbide would progress
rapidly to completion, provided only that free exit were allowed for the gaseous
products of reaction. It must, however, be remembered that the process is
eudothermic, and can therefore only proceed at a pace measui-ed in terms of the rate
at which energy is being supplied to the furnace.

We have carried out a large number of experiments specially to study the effect
of the presence of carbon monoxide upon the yield (see Table III.).

In these and in other cases the resulting product was submitted to a careful
examination and analysis.

The sectional view shown in fig. 11, AI, gives an idea of the general appearance
of the furnace after the run.

The furnace contents consist of (a) ingot of fused calcium carbide, (6) fritted mass
surrounding the central cavity, (c) residual unacted-on material. These were
separately collected and weighed, then parted and sampled, and subsequently
analysed in the manner described below.

Generally speaking, the central lump represented the entire yield of carbide,
although small quantities of acetylene were sometimes obtained from the fritted mass.

The yields recorded are all calculated from the amount of acetylene produced. The
gas evolved was always carefully analysed, as the possibility presented itself of the
formation of other carbides or free calcium metal. The amount of impurity was,
however, invariably found to be insignificant. The outside unfused material was
examined, but gave no appreciable evolution of a combustible gas when acted upon
by water, or even by dilute hydrochloric acid.

The ingots of carbide showed a good crystalline fracture. The purity of the lump
was, as might be expected, below that of a good grade technical product, but
increased as the rate of power expenditure rose, and, curiously enough, was entirely
independent of the presence or absence of carbon monoxide.

When the carbon monoxide was retained in the furnace the ingot frequently
showed on its upper surface a thin coating of bright graphite, giving it a metallic
appearance, and in some few cases narrow strata of graphite plates occurred within
the mass itself.

* A. FRANK, ' Z. fur angew. Chemie,' 1905, vol. 18, p. 1733.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

441

The experiment detailed in Table IV. was one of those in which all the gaseous
products of reaction were retained in the furnace until long after the completion
of the run.

Fig. 12, which refers to this experiment, is reproduced, since it is typical of many
of those dealt with in the section.

10 20 ^0 40

TlHC IH HIMUTtS

SO 60

Fig. 12. Energy, power, and pressure curves of a calcium carbide experiment.

The curves refer to the experiment dealt with in Table IV., in which the furnace was started at
atmospheric pressure and all the gas generated was retained.

Curve A represents the power in kilowatts at each instant.

Curve B is obtained by integration of A, and gives therefore the total energy expended up to
any given time.

The readings of the pressure gauge on the furnace, taken at intervals, are marked on the
diagram, and fall nearly upon the energy curve.

The power at each instant during the run is given in curve A, whilst the total
energy expended up to any given moment, obtained by an integration of this, is
given at B.

In all the experiments the pressure after the first few minutes rose at an almost
constant rate. When the pressure oljservations are plotted to a suitable scale they
coincide with the integrated power curve. In the figure their position is indicated,
and the concordance can thus be clearly seen.

The interpretation is not so simple as it appears, for there are two disturbing

VOL. CCVIL — A. 3 L,

442 MESSRS. R. S. HUTTON AND .T. E. PETAVEL ON ELECTRIC

factors which must be taken into account : firstly, the thermal loss, which increases as
the hot zone widens out, and, secondly, the variation of the average temperature
of the furnace, the effect of which will superpose itself upon the actual gas evolution
and thus augment the pressure readings. Apparently these two factors counter-
balance each other.

From the analysis of the furnace gas [see Table IV. (3)] a re-absorption of carbon
monoxide is clearly indicated. The fall of pressure after the end of the run cannot
of course form the criterion, as its amount will vary with the distribution of
temperature in the furnace. The considerable decrease in the percentage of the
carbon monoxide can, however, only be ascribed to re-absorption, and the results are
thus so far in agreement with those of the observers mentioned above.

That this absorption is not due entirely to physical causes is shown clearly by
comparison with a similar series of analyses carried out in the case of carborundum
(Table VII).

There is little doubt that the graphitised surface of the ingot, referred to in a
preceding paragraph, is a result of the recombination of the carbon monoxide. The
instant the power is cut off, the temperature of the molten mass begins to fall rapidly,
and it is during this period that the back reaction chiefly occurs. As soon as the
product has frozen, further attack is limited, for the ingot of carbide is of a very
compact and impermeable texture, and is protected by the graphite film.

Such a " skin reaction " would doubtless become predominant were the experiments
carried out on a few grammes of material, but when dealing, as we are here, with
larger quantities, the total loss is too small to influence appreciably the result. This
is shown by the fact that the yield is not increased when the carbon monoxide is let
off immediately upon stopping (cf. Table III., C 26).

The further question, as to whether the presence of an atmosphere of carbon
monoxide during the run has an unfavourable effect on the efficiency of the process,
is also answered in the negative by a comparison with furnaces operated at
atmospheric pressure, but otherwise under identical conditions.

These unexpected results called for more detailed study, and a number of
experiments were undertaken in which provision was made for carrying away the
carbon monoxide as soon as it was formed. The first method which suggested itself
was to remove the gaseous products of reaction by dilution with some inert gas,
which was alternately introduced and discharged, the pressure in the furnace being
made to fluctuate between two fixed values. The record of such an experiment
with coal gas will be found in Table III. (C 22), showing, if anything, a decreased
yield.

A more efficient method of washing out the carbon monoxide was then devised.
A hollow carbon electrode was brought into use, and during the entire run a constant
stream of pure hydrogen was injected directly into the reaction zone of the furnace.
The current of gas was also maintained during the cooling, the quantity of gas

ftTUNACK REACTIONS UNDER HKili CASEOUS PRESSURES. II.;

employed in each experiment being some 2000 litres. The pressure in the furnace
was regulated and kept constant by one of the valves placed on the side of the
enclosure, through which the required amount of gas was allowed to escape.

In view of these experiments, arrangements had been made for communication
between those engaged in operating the furnace and the worker in charge of the
compressor. The gas connections were so disposed that the gas could be delivered
either directly into the furnace or into a receiver communicating therewith. Gauges
in the furnace room indicated the pressure on the pump, and also the working
pressure of the enclosure.

Most gases had of course to be simultaneously manufactured and compressed, but
coal gas, drawn directly from the mains, was occasionally used for simply washing
out of the products of reaction.

The result of the circulation was not to increase, but considerably to decrease the
efficiency of the carbide formation. It occurred to us, however, that the low yields
might be ascribed to the thermal losses entailed by the specific heat and the relatively
high conductivity of the hydrogen used for dilution.

In order to be quite free from such objections, it was decided to remove the carbon
monoxide as fast as it was formed by means of a pump, and carry out the reaction
under a partial vacuum.

To protect the pump from the large quantities of finely divided material, which
are carried away with the stream of gas, a number of scrubbers and filters were used.
When the furnace is operated at full power, the gas generated by the reaction
amounts to some 30 litres per minute, and, although an exceptionally powerful
vacuum pump was available, it was only possible to maintain the vacuum at about
30 to 40 centims. of mercury. The average yield obtained in the vacuum experiments
does not materially differ from the results already given.

We are therefore justified in concluding that, however contradictory it may seem,
even a concentrated and compressed atmosphere of carbon monoxide has no dele-
terious effect upon the formation of calcium carbide.

Having entered so fully into the important question of the influence of carbon
monoxide, it is necessary to deal very briefly with other sides of the question.

Generally speaking, within wide limits (between 5 and 20 kilowatt hours) the
total power consumption does not affect the efficiency of the process.

The influence of pressure per se has not resulted in any marked change in the
chemical or physical nature of the products, neither can a considerable decrease in the
yield be traced to this cause. Such variations in the purity or richness of the carbide
as have been noticed are attributable only indirectly to pressure, being accounted for
by the increased thermal losses in high pressure gases.

Finally, we hope that the general methods of following the course of the reaction
by a measurement find analysis of the gaseous products will be as useful when applied
to other problems as they have been in this special case.

3 L 2

444 MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

ON THE FUSION or SILICA.

When quartz is subjected to a high temperature it attains the vitreous condition
and becomes sufficiently plastic to be moulded and shaped.

It was shown some years ago that silica can be heated in direct contact with carbon
and brought to a plastic state without marked reduction occurring.

The heating was effected either by radiation from an electric arc or by placing the
material around a carbon core* through which the current was passed.

At ordinary pressures it is, however, impossible to maintain any considerable
quantity of silica in a really liquid state.

When the power expenditure in the core is increased a rapid volatilisation sets in
which effectually puts a limit to the temperature.

It seemed probable that more satisfactory results could be attained by carrying out
the fusion under a considerable pressure in the enclosure described above.

Details of the principal experiments will be found in Table V. A resistance
furnace of construction similar to that shown in fig. 11, B, was first employed, the
central core of granular carbon being replaced by a carbon tube held in two massive
graphite terminal pieces to which the current was led. This core was arranged
centrally and surrounded by pure quartz sand, the experiments being carried out in
air at 50 and 100 atmospheres.

Thick-walled hollow cylinders, 25 centims. long and 15 or 20 centims. external
diameter, were in this way easily obtained.

At first sight more complete liquefaction seemed to have occurred. Upon fracture,
however, the material was found still to contain innumerable small gas bubbles,
giving it a translucent appearance and tending to show that the fluidity had not been
much increased.

Proof of the diminished volatilisation of the material was, however, given by
the absence of a deposit of condensed silica vapour, as also by the very small
formation upon the core of carborundum, both of which are evident at atmospheric
pressure.

It was then decided to study two modifications of the regime, either of which
seemed likely to give improved results.

The well-known ease with which hydrogen passes through heated silica led us to
believe that if a compressed atmosphere of this gas were employed any bubbles
imprisoned at the moment of fusion would disappear, leaving the glass clear. None
of our experiments, however, verified this assumption. Not only is the occlusion of
the gas apparently unaffected, but from the nearly explosive violence with which the

* HUTTOX, 'Mem. Manch. Lit. and Phil. Soc.,' 1901, vol. 46, No. 6, pp. 1-5; also 'Trans. Amer.
Electrochem. Soc.,' 1902, vol. 2, pp. 105-111.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES. 445

material splits when subjected to shock there is little doubt that each minute cavity
still retains gas under a considerable pressure.*

The small weight of vitrified product is ascribed to the high cooling effect of
hydrogen, but the properties of the material were similar to those noted in the earlier
work.

Finally, an attempt was made to attain the highest {>o88ible temperature by
concentrating the heat of a powerful arc in the centre of a mass of sand, the
equipment of the furnace being of the type shown in fig. 11, D.

No difficulty was experienced in maintaining an arc of some 10 kilowatts for an
hour or more, the electromotive force rising to 120 volts under a pressure of 180
atmospheres.

A hollow sphere of 18 centims. external diameter and weighing about 5 kilogs.
was obtained, which, however, was no more transparent than with the less intense
heating.

ON THE FORMATION OF CARBORUNDUM.

In 1893 ACHESON discovered that a mixture of carbon and silica heated around a
resistance core readily produces a carbide of silicon to which the name of carbo-
rundum was given. Its preparation is now carried out on a large scale. It was
therefore of some interest to study the reaction, first under the usual conditions, then
under high pressure.

We have prepared some quantity of this material in the laboratory and found it to
differ from the commercial product only in the smaller size of the crystals. The
amorphous variety invariably occurs in considerable amount surrounding the crystal-
line layers.t

In carrying out the reaction in the pressure furnace the mixture of sand and carbon
was disposed around a core of granular retort carbon as shown in fig. 11, B.

To insure a uniform cross-section this resistance core was tightly packed into a
thin brass tube which was placed in the axis of the furnace and supported between
the two graphite electrodes. As the mixture when heated becomes somewhat
conducting, it is advisable to line the furnace with a thin insulating layer of sand or
other material in order to avoid a short circuit through the iron.

In these experiments the gas generated by the reaction was retained in the furnace
and the pressure allowed to accumulate up to some fixed limit (100 atmospheres).

A typical example of such a run is given in Table VI. As will be seen, the brass

* Relative to this subject an interesting investigation has been carried out by ARTHUR L. DAY in
America. Using lower pressures, which, however, were only applied after a high temperature had been
attained, he has prepared a glass containing relatively few air bubbles. ' Science,' N.S., 1906, vol. 23,
pp. 670-672.

t See also S. A. TUCKER and A. LAMPEX. ' Journ. Amer. Chem. So<.-.,' 1906, vol. 28, pp. 863-868.

446 MESSRS. R S. BUTTON AND J. E. PETAVKF. OX KLECTRIC

tube fuses within the first few minutes ; the power can then be rapidly increased, arid
the temperature of formation of carborundum is soon attained.

There is little doubt that the production of carborundum is the result of the
interaction of the vapour of silica and the highly heated granular carbon. From the
previous work on quartz we were therefore led to anticipate that under pressure the
reaction would not occur very readily.

Several experiments confirmed this impression. An examination of the furnace
after the run showed, in every case, that, though the mixture of sand and carbon had
attained a sufficiently high temperature to effect the fusion of the quartz to a
considerable depth, thus agglomerating the mixture, only a small quantity of
carborundum was formed and that immediately around the central core.

Another distinctive feature of the pressure experiments is the almost entire absence
of the amorphous variety of carborundum. The reaction progresses at a slow rate,
but apparently uniformly, the resulting pressure being, as in the case of calcium
carbide, a linear function of the time (see fig. 12).

In order to study more in detail the progress of the reaction, an experiment was
planned in which the whole of the gas generated was retained in the furnace.

As is shown in Table VII., analyses were made at intervals both during the run
and in the subsequent cooling period. From these it is clear that in this case there
is no inverse reaction. The high absorbing power of carbon for carbon monoxide as
compared with hydrogen fully accounts for the slight decrease in the percentage of
the former.*

ON THE DIRECT REDUCTION OF ALUMINA BY CARBON.

The methods used in practice for obtaining aluminium from its ores are indirect and
inefficient.

The preparation involves a lengthy and complicated purification of the oxide,
followed by its electrolysis in a bath of cryolite. Early work showed that where
aluminium alloys are required they could be obtained by a simple method involving
the reduction of alumina by carbon, but the process has never been successful for
the production of the pure metal. Up to the present time opinion seems to be
divided as to the effect of heating alumina and carbon together in the electric
furnace.

Several authorities definitely state that alumina is irreducible by carbon.t whilst
others affirm that it is quite easily reduced.^

* DEWAK, 'Roy. Soc, Proc.,' 1904, vol. 74, pp. 122-127.

t W. HAMPE, ' Chemiker-Zeitung,' 1888, 12, 391. S. A. TUCKER and H. R. MOODY, 'Journ. Soc.
Chem. Ind.,' 1901, 20, 970.

J COWLKS see W .P. THOMPSON, 'Journ. Soc. Chem. Ind.,' 1886, vol. 5, p. -'06; W. BOKCHERS, 'Elektro-
Metallurgie,' 3tc Aufl., 1903, p. 102.

I IRNACE REACTION^ [ \DER HKJH CASEOUS PRESSURES. 447

MOIHHAN,* taking an intermediate position, asserts that the two materials only react
when in the form of vapour.

The question, which for many reasons is of considerable importance, has never
received the detailed investigation which it deserves.

Our experiments at atmospheric pressure, as we shall see, pointed to the fact that
a well-marked thermal reaction does take place, but not until the fusing point of
alumina is reached.

HKROULT,! while admitting that reduction occurs, attributes it to electrolytic
action. Having carried out some experiments in which the reacting sulxstances were
In -a ted by radiation alone and in which good yields of aluminium bronze were
obtained, we contend that the assumption of electrolysis is by no means necessary.

No information was available as to the temperature of vaporisation of metallic
aluminium, but various observations led us to believe that a large proportion of the
reduced metal was lost by volatilisation and subsequent combustion where the
furnace gases come in contact with the air.

The high-pressure furnace seemed to us therefore particularly suitable for studying
this question, the advantages to be gained consisting firstly in the complete protec-
tion of the products from oxidation, and secondly in the decreased volatilisation which
might be expected under the high gaseous pressures.

Some of the experiments tried under pressure to study this problem are given in
Table VIII., details of one experiment being reproduced in Table IX.

By a cursory inspection of Table VIII. the two following facts may at once be
deduced :—

(1) That in the resistance furnace neither aluminium nor its carbide is produced.

(2) That on the other hand all arc furnaces give a more or less marked reduction ;
although it will lie noticed (in section B) that the product chiefly occurs as carbide of
aluminium.

In several cases small malleable lumps of the metal were condensed in the powdered
material surrounding the fused product.

From this it would appear that the required conditions for which we are searching
had for some short period t>een accidentally fulfilled — these conditions being the
rapid removal of the metal vapour from the reduction zone and its condensation
under circumstances which precluded carburisation.

The idea that, by reducing the partial pressure of the carbon monoxide by
a circulation of hydrogen or coal gas, more favourable results would be attained led
to the experiments quoted in Table VIII., 0 and D. From these we infer that the
reaction is considerably favoured by a dilution of the carbon monoxide. It is further
noticeable that this precaution results in an increase in the relative quantity of

* H. MOISSAN, < The Electric Furnace,' London Ed., ARNOLD, p. 184.

t P. L. T. HEROUI.T, 'Eng. Pat. 16853,' 18*7; alqg 'Congrf-s intern, des Mines et de la Me"tallurgie '
(Paris), 1900.

448 MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

aluminium metal, although it is still accompanied hy a considerahle amount of the
carbide.

It therefore became evident that further work at high pressures must be preceded
by a more detailed study of the conditions of reduction. The several problems which
arise may briefly be stated as follows :—

(1) At what temperature does alumina first show signs of reduction by carbon ?

(2) In the production of aluminium alloys, what is the function of the auxiliary
metal in facilitating the reduction of alumina ?

(3) What precautions are necessary to limit the formation of carbide and increase
the production of metal ?

Since it is well known that alumina cannot be reduced under ordinary circumstances
in the Moissan furnace, it was thought advisable to see whether, by carrying out the
reaction in an atmosphere of hydrogen, a definite indication of reduction could be
obtained. The Moissan furnace was, of course, modified to exclude the use of limestone
and the accompanying production of carbon monoxide.

As will be seen from Table X., A, a negative result was obtained.

As a means of limiting the temperature of reaction, calcium fluoride was introduced,
but no signs of reduction were apparent at the boiling-point of the bath. From these
and similar negative results at lower temperatures, which it is unnecessary to record,
we assumed as a working hypothesis that the temperature of reduction of alumina is
above the boiling-point of aluminium metal under atmospheric pressure.

The hypothesis we confirmed by experiments (Table X., B) in which special
precautions were taken to protect the material from access of air and to provide a
condensing chamber in which the vapours were cooled down before their exit from
the furnace. The deposit so obtained showed unmistakeable evidence of the presence
of finely divided aluminium.*

It therefore became necessary to devise some better means for indicating the
production of any metal vapour.

A method which suggested itself to us, and one which has proved of considerable
usefulness, was the employment of a bath of molten copper, on the surface of which
the reaction mixture was placed. The copper served as an absorbent for any
aluminium vapour liberated.

To determine the lowest temperature at which reduction occurs, a series of
experiments was carried out. Small carbon crucibles were used to contain the mixture.
These were heated either in a carbon tube furnace, or, for higher temperatures, more
conveniently by embedding them in a granular carbon resistance. From the summary
of these experiments in Table X., C, it will be seen that the minimum temperature of
reduction coincides fairly sharply with the melting-point of alumina, and is not

appreciably lowered by the introduction of either fluor spar or lime as a flux.

•
* See also C, F, MAPERY, ' Amer. Chem. Journ.,' 1887, vol. 9, pp. 11-15.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES. 449

By referring again to Table VIII. it will be found that this view is substantiated
by a comparison between the arc and resistance experiments.

In the latter the yield is always extremely low. This may be explained by the
fact that as the inner layer of mixture approaches its fusing point it flows away by
gravity, and, ceasing to transmit the current, is not maintained at the requisite
temperature for marked reduction to occur.

We come now to that curious apparent contradiction of facts which has for so long
puzzled investigators in this field ; namely, that though aluminium bronze and ferro-
aluminium can be so readily produced, no process exists by which the metal itself can
be obtained from the oxide, except by indirect means. What is, then, the function of
the auxiliary metal ? It has been suggested that a marked chemical affinity exists
between the aluminium and the metal with which it alloys, the evidence in support
of this being the high heat evolution which is noticeable when aluminium is added to
the metals in a molten state.

It must, however, be remembered that under ordinary conditions the fused metals
contain dissolved oxide, and it therefore seemed worth while to carry out a preliminary
investigation of this question.

Upon adding aluminium to molten copper in a thoroughly reduced condition, there
is no visible evidence of a reaction, and such pyrometric measurements as were made
sufficed to show that no considerable amount of heat could have been evolved.

Thus we feel justified in concluding that the copper or other metal serves chiefly
to condense and dissolve the aluminium, and does not itself take part in the primary
chemical reduction of the oxide.

A secondary function of the auxiliary metal is, however, possible. It occurred to
us that the absence of aluminium carbide, when reduction is effected in the presence
of other metals, might be explained by some chemical action of the aluminium carbide
upon the copper or iron or one of their oxides.

An investigation of this matter has been undertaken by J. N. PRING,* whose
results clearly show that at the temperatures we are considering, namely, at or above
the melting-point of alumina, aluminium carbide reacts with either the oxide or the
metal, forming an alloy.

The third problem, viz,, the limitation of the formation of carbide, seems to be the
most difficult to solve.

As we have seen, the metal may be considered to exist in the form of vapour at the
moment of its reduction. Owing to the well-known affinity of aluminium for carbon
monoxide, t it is obviously important to remove this gas as completely and rapidly as
possible.

A method of reducing the partial pressure of the carbon monoxide has been dealt
with above, and we have found it important to lead the gas used for dilution directly

* J. N. PRING, -Trans. Chem. Soc.,' 1905, vol. 87, p. 1530.
t GUNTZ and MASSON, 'Comptes Rendua,' 1897, vol. 124, p. 187.
VOL. CCVII. — A. 3 M

450 MKSSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

to the seat of reaction by means of a hollow electrode, the stream of gas thus not only
effectively diluting the carlxm monoxide, but serving to carry forward the metallic
vapour into a zone more favourable for its condensation.

Even in the absence of carbon monoxide, carburisation can occur by direct union
of the metal with solid carbon.

Some unpublished work of W. H. PATTERSON carried out in this laboratory has,
however, shown that in the absence of carbon monoxide this reaction only occurs
above a bright red heat, thus explaining the results already quoted in which the
metal was obtained, although doubtless it had not altogether escaped contact with
carbon.

We are therefore in the following position : we have proved the facility with which
the direct reduction of alumina by carbon can be effected, and have shown that the
minimum temperature at which it can occur is already sufficiently high for the metal
to be produced in the form of vapour.

Future work must be directed towards the application of high pressure for reducing
the vaporisation of the metal at the temperature of reaction, the rapid removal or
dilution of the carbon monoxide by a stream of inert or reducing gas, and a modifica-
tion of the regime to facilitate the condensation and prevent the collected metal from
flowing into a bed of highly heated carbon.

Thus the necessary conditions for the successful direct reduction of alumina by
carbon seem to be fairly well defined, the outstanding problem being chiefly a matter
of the arrangement and construction of the furnace.

The cost of most of the apparatus, which was specially constructed for the above
research, was defrayed by funds awarded by the Government Grant Committee of the
Royal Society. We have also been materially assisted, so far as the gas preparation
plant is concerned, by Messrs. Brunner, Mond & Co., Ltd., and the Tudor
Accumulator Co., Ltd.

With regard to running expenses, the work has been much facilitated by the kind
way in which the ample resources of the Physical Laboratory of the Manchester
University have been placed at our disposal.

In concluding, we desire to express our heart-felt gratitude to Professor ARTHUR
SCHUSTER for the never-failing interest and encouragement which he has given us
during the several years over which the research has extended.

FURNACE REACTIONS UNDER IIIOH GASEOUS PRESSURES. 451

TABLE I. — E.M.F. and Current of Carbon Arc under Pressure (see fig. 10).

A.

In air at atmospheric pressure.

+ ve 41 millims.

- ve 27 mi Him-.

Length of arc 8 millims. *

iuii|H-n-i TolU

40 60]

80 5

118 46 i- silent.

180 63

250 66

B.

In rarln m monoxide.

Both carbons 27 millims.

Length of arc 1 to 2 millims.

(1) At 12 atmospheres.

ampere* roll*

38 42

60 32

140 27

(2) At about 90 atmospheres.

30 140

100 110

150 115

C.

In nitrogen.

Both carbons 27 millions.
Length of arc about 2 millims.

(1) At 1 3 atmospheres.

!IIII|H'TI'< TolU

50 60

55 58

60 58

65 50

(2) About 20 atmospheres.
60 120'

60 124

D.

In carbon monoxide.

Both carbons 1 1 millims.

Length of arc 1 to 2 millims.

(1) At 11 atmospheres.

70
75
80
80

120
117
117
120

iinipi-n-4

12
13
15
25

27
38
48

TolU

70
68
61
72
70
79
82

abnormal
running.

(2) At 16 atmospheres.
32 110

36 108

44 120

60 128

E.

In air.

+ ve carbon 4 1 millims.

- ve carbon 27 millims.

Length of arc about 2 millims.

(1) Pressure, atmospheric. <2> A* 15° "^spheres.

""uT TK^ """>"' 130*

30 41\ 16 110

50 37 ri88in«- 17
80 40J

20
40

90
82
68

(3) At 190 atmospheres.
•mp£re« Toltt

90 120'

95 120

100 108 f mnning-

abnormal

* In A only the length of the arc is the distance between the point of the negative of the edge of the
crater of the positive. In all other experiments it is the distance of feed required to produce actual
contact between the two electrodes. Solid carbons were used throughout.

3 M 2

452

MESSRS. R. S. BUTTON AND J. E. PETAVEL ON ELECTRIC

TABLE II. — Oxidation of Metals in Air under High Pressures.

Experiment No.

A 1, 3, and 4

A5

A6

A7

A 13

A 12

Crucible filled with copper, iron, or aluminium, heated for half-an-hour in furnace of
type fig. 11, C, under carbon arc. Pressures, about 30 atmospheres. Power, 5
to 10 kilowatts. Metals fused, but the oxidation as shown by the gas analysis
was practically limited to the carbon.

Arc between iron electrodes. Pressures, 27 to 75 atmospheres. Metal at end of
electrode apparently violently boiling, but after more than an hour still over
20 per cent, oxygen. Subsequently iron bar maintained at bright red heat by
current of 1000 amperes for about 2 hours without entering into combustion.
Oxygen at end of run over 20 per cent.

Iron electrodes, 1TV inches diameter, in oxygen at 15 atmospheres. Points of
electrodes heated by 300-ampere arc. Vivid combustion started. Current at
once cut off. Positive electrode continued burning until pressure was reduced to
5 atmospheres. Total iron burnt, 550 grammes.

Rod of iron, £ inch diameter, 1J inches long, between iron electrodes in air at
100 atmospheres, maintained at bright red heat by powerful current and then
fused without starting combustion, Over 20 per cent, oxygen after run.

High-tension arc (1000 volts) between thin iron rods.
Combustion not started.

Arc between copper electrodes at 92 atmospheres,
oxygen.

Pressure, 110 atmospheres.

After run, 20-5 per cent.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

i:..;

TABLE III. — Summary of the Principal Experiments on the Production of

Calcium Carbide.

Experi-
ment
No.

Total

energy.

Average
power.

Initial
pressure.

Maximum
pressure.

Product,
grammes,
CaC,
from
analysis.

Yield,

grammes
per
kilowatt
hour.

Observations.

kilowatt

hours

kilowatt*

atmosphere*

atmosphere!

C29

3-84

11-5

Atmospheric

30

283

73-7

CO from reaction retained

in furnace.

C27

6-2

12-0

Idem

50

504

81-4

Idem.

C 58

9-26

13-2

Idem

42

703

75-9

Idem.

C61

12-6

14-5

Idem

120

1088

86-7

Idem.

C23

15-0

15-8

Idem

15

1258

84

Pressure kept at 1 5 atmos-

pheres, excess of CO

being let off.

C26

6-5

13-9

Idem

48

391

60-2

CO retained during ran,

but let off at once on

stopping.

C22

14-9

14-2

10 coal giis

25

1017

68-3

Circulation of coal gas by

"

alternately admitting

and blowing off gas

from furnace enclosure.

Total coal gas, 1420

litres.

C65

9-75

12-7

50 hydrogen

50

405

41-5

Circulation of hydrogen

through hollow carbon

electrode. Total hy-

drogen, 1705 litres.

C71

12-4

9-78

50 hydrogen

59

238

19-2

Idem. Total hydrogen,

2420 litres.

C66

6-48

11-7

Atmospheric

Minimum

455

70-3

Vacuum. Maintained

30 centims.

below £ atmosphere

Hg absolute

during entire run.

C67

15-8

17-2

Idem

Minimum

1385

87-7

Vacuum. Idem.

35 centims.

Hg absolute

C73

13-5

13-5

Idem

Atmospheric

1437

106-4

Atmospheric pressure.

454

MESSRS. R. S. HUTTON AND J. E. PETAVEL ON ELECTRIC

TABLE IV. (1). — Details of one Typical Calcium Carbide Experiment (C. 61).

The large furnace enclosure was used in the vertical position (see Plate 6, fig. 1).

Iron liner filled as shown in fig. 11, A.

Lower electrode ( - ve), bed of 2 kilogs. retort carbon.

Upper electrode ( + ve), a 41-millim. carbon rod, 32 centims. long.

Weight with holder, 1657 grammes. Loss of weight during run, less than 5 grammes.

Charge : alxmt 8600 grammes of a mixture of selected Buxton lime and petroleum coke (100 CaO : 65 C).
Lime all passed through a 20-mesh sieve, petroleum coke 60-mesh. The vessel with its contents was
heated for some hours to a red heat before the experiment to ensure complete drying of material.

Initial pressure, atmospheric.

The gas capacity of the furnace when charged was found to be 1 1 • 6 litres per atmosphere.

All the gaseous products of reaction were retained during the experiment and for 42 hours after
completion of same, with the exception of the small quantities required for gas analysis at intervals as
recorded below.

The E.M.F. at the terminals of the generators was varied as required and was usually 5 to 10 volts
above that on the furnace.

TABLE IV. (2).

Time.

Current.

E.M.F.

Power.

Pressure.

1
Observations.

minutes

amperes

volts

kilowatts

atmospheres

start

70

27

1-89

—

2

200

21

4-20

—

4

300

30

9-00

—

6 360

48

17-3

2

7 380

42

16-0

5

12

400

41

16-4

17

1st sample of gas taken for analysis.

15

350

48

16-8

30

17

140

70

9-8

38

22

240

63

15-1

46

23

300

50

15-0

48

2nd sample of gas taken for analysis.

25

360

40

14-4

55

27

340

47

16-0

60

32

300

54

16-2

70

3rd sample of gas taken for analysis.

34

280

58

16-2

75

37

300

58

17-4

84

43

280

58

16-2

96

44

240

67

16-1

103

4th sample of gas taken for analysis.

50

220

69

15-2

113

52

200

58

11-6

120

5th sample of gas taken for analysis.

stop

—

—

—

—

Temperature of outside of the main

«

enclosure at stop, 40° C.*

* At 30 minutes after stop the cover attained a maximum temperature of 65° C., falling in

2 hours to 30°.

FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

455

TABLE IV. (3). — Record of Analyses of Furnace Gas.

Sampl
No.

e

Time after stop.

Furnace
pressure.

C02.

CO.

H.

CH4.

N.

1

atmosphere*
f 1?

per cent.

1-6

per cent.
85-1

per cent.
8-2

per cent.
1-1

per cent.
4-0

2
3
4
B

1 tuning run, see
previous table

48

1 70
I 103
[120

1-2

0-5
0-3
1-1

85-3
86-5
89-0
89-6

7-4

13-5
13-0
10-7
0-9

1-0

6
7
8
9

0 8
0 25
0 39
0 58

92
55
43
34

3-4
1-7

2-7
1-3

85-3
80-1
77-9
76-1

14-8

11-3
18-2
19-4
4-9

2-9

10
11
12

1 13
2 43
3 40

30-5
21-5
18-5

2-0
1-5.
1-8

74-0
71-1
69-1

16-7
19-1

24-0
6-2
7-2

4-5

2-8

13
15

22 0
42 0

15
12

1-4
1-4

66-9
65-6

20-2

31-7
8-3

4-5

In this, as in most other cases, the furnace gas was found to contain a small percentage of

iron carbonyl.

TABLE IV. (4).

The appearance of the furnace after run was similar to that shown in fig. 11, A;.

The products from the different zones of the furnace were collected separately and submitted to analysis
with the following results : —

Gross weight.

Weight of CaCj from analysis.

Fused lump

gramme*
2090

gramme*

1088

Fritted mass round lump . .
Fritted mass from base . . .
Unacted-on mixture ....

1030
770
6570

negligible
ii
ii

456

MESSRS. R. S. BUTTON AND J. E. PETAVEL ON ELECTRIC

TABLE V. — Summary of the Principal Experiments on the Fusion of Quartz.

Experi-
ment
No.

Type of furnace.

Total
energy.

Average
power.

Gas.

Working
pressure.

Observations.

-

kilowatt
hours

kilowatts

atmospheres

D62

Resistance. Carbon

8-34

10-0

Air

100

Product, a tube 23 centims.

tube, 4 centims.

long, 4 '5 centims. in-

external diameter,

ternal diameter, quite

. length, 22 centims.,

detached from core. Ex-

between graphite

ternal diameter about

terminals

8 centims. Weight of

vitrified product, 2000

grammes.

D68

Resistance. Carbon

12-9

9-94

Air

100

Weight of vitrified pro-

tube, 3 centims.

duct, 4100 grammes, for

external diameter,

greater part of its length

length, 21-7 cen-

adhering to carbon core.

tims., between

At top blown out, forming

graphite terminals

a cup about 10 centims.

internal diameter.

D72

Resistance, as in D 68

11-3

9-14

Hydrogen

50

Weight of vitrified product,

2100 grammes, moulded

around core.

D74

Arc. 100 amperes at

9-52

8-94

Hydrogen

180

Hollow sphere of vitrified

120 volts, between

material, 18 centims.

carbon electrodes

external diameter, weigh-

(fig. 11, D)

ing 4900 grammes.

iri;\.\(T.

men <:ASK<>IS

457

TABLE VI. — Details of One Typical Carborundum Experiment.

Large furnace enclosure used in vertical position (Plate 6, fig. 1).

Iron liner filled as shown in fig. 11, B.

Central resistance core of granular retort carbon packed in thin-walled brass tube 30 centims. long,
i''ii ri-iiiiiiis. diameter.

Electrical connection at top and bottom of core made by graphite discs 5 centims. thick and 8 centims.
diameter.

Charge : mixture of 7J kilogs. white Calais sand and 4 J kilogs. finely ground retort carbon ; calcined
shortly before experiment.

Iron pot lined with thin sheet of asbestos.

Initial pressure atmospheric; gaseous products of reaction retained up to about 100 atmospheres, then
let off and pressure maintained constant.

E.M.F. at the terminals of the generator varied as required, and usually 2 to 3 volts above that in the
furnace.

Time.

Current.

E.M.F. on
furnace.

Power.

Pressure.

Observations.

inimile-

unp^m

rolu

kilowatt*

atmospheres

start

220

4-0

0-88

Atmospheric

1

480

7-0

3-36

—

3

600

7-0

4-20

—

8

600

6-0

3-60

—

W

840

8-0

6-72

—

13

1000

8-5

8-50

—

14

500

12-0

6-00

—

Sudden increase of resistance due

to fusion of thin brass tube.

16

550

22-0

12-1

1

18

530

29-0

15-4

8

19

410

38-0

15-6

13

21

520

31-0

16-1

20

22

520

29-5

15-3

26

23

520

29-5

15-3

33

25

500

30-5

15-2

43

27

470

32-5

15-3

52

30

370

40-5

15-0

71

34

300

46-0

13-5

93

37

340

41-0

13-9

108

Started letting gas off into gaso-

meter.

38

360

41-0

14-8

__

39

380

40-0

15-2

—

41

410

38-0

15-6

107

56 litres had been collected in

gasometer.

44

440

36-0

15-8

105

269 litres had been collected in

gasometer.

46

470

35-0

16-4

101

49

470

33-0

15-5

108

52

470

33-0

15-5

107

55

470

33-0

15-5

102

VOL. CCVII. — A.

3 N

458

MESSRS. R. S. BUTTON AND J. E. PETAVEL ON ELECTRIC

TABLE VI. — Details of One Typical Carborundum Experiment (continued).

Time.

Current.

E.M.F. on
furnace.

Power.

Pressure.

Observations.

minutes

amperes

volts

kilowatts

atmospheres

58

490

32-0

15-7

106

62

490

32-0

15-7

100

65

490

32-0

15-7

—

67

480

32-0

15-4

104

70

480

32-5

15-6

73

480

32-5

15-6

103

76

480

32-5

15-6

103

Stop.

Total gas production during the run calculated to be 840 litres at 0° C. and 760 millims.

Product closely adhering to central core formed a fritted mass cylindrical in shape, about 15 centims.
diameter and 27 centims. long.

Total weight, 3600 grammes. Inner layer about 2 centims. thick, consisting of crystalline carborundum
surrounded by thin sheath of the amorphous variety. The outer layers on analysis found to consist of
agglomerated carbon and sand, containing only small percentage of carborundum.

Granular carbon core had been graphitised, but contained no carborundum.

TABLE VII. — Record of Furnace Gas Analyses in a Carborundum Experiment.

Sample
No.

Time.

Furnace
pressure.

OQ>

e

CO.

H.

CH4.

N.

minutes

atmospheres

per cent.

per cent.

per cent.

per cent.

per cent.

1

From start, 21

26

5-2

70-0

24-8

—

2

34

55

3-7

83-3

—

13-0

—

4

62 (stop)

122

2-9

89-0

4-2

0-7

3-2

5

After stop, 5

120

4-9

86-2

—

8-9

—

6

20

94

4-5

85-6

—

9-9

—

7

36

80

5-7

83-8

—

10-5

—

8

61

67

6-4

82-7

—

10-9

—

9

154

46

5-4

83-5

6-3

1-2

3^6

FURNACE KKACTIONS UNDER HIGH GASEOUS PRESSURES.

ii

f !

e

fl

'S
co

Atmospheric

Atmospheric

Atmospheric

c

1

•S

co

55 hydrogen

I
1

8

QO

lO

1

1

1

o

»— 1

3 hydrogen

a

T3 •

-=

§

e
•S

S

Ik

3

I *

O

0

00

C-J

o

t—

CM

0

«

00

0

00

o

* o

o>

co

**

00

co

s

t-
1-1

co

CO

0

00

0

t-

3 g

^ o

•0

(M

„,

CO

o

„

<M

0

co

M

0

eo

„

- :

00 CO

«

CO

at

-*

0

1—1

1— 1

t-

30

2

•

CM

0

i— i

CO

•

, 6

t- «^

Oi

r^

<•

t~

o

00

t-

O

^H

•<*•

eo

en

10

co

$ A3

t-H

^1

H*

r- 1

CO

t—

e-

C4

•i

CO

CO

i-

t—

0

O

0

o

o

o

0

O

o

0

o

O

O

0

S i

3 N 2

4(50

MESSRS. R. S. BUTTON AND J. E. PETAVEL ON ELECTRIC

TABLE IX. (1). — Typical Experiment on the Reduction of Alumina by Carbon (G 33).

The large furnace enclosure was used in the vertical position.

Iron liner filled as shown in fig. 11, A.

Lower electrode, bed of 2 kilogs. retort carbon.

Upper electrode, a 41-millim. carbon rod 32 centims. long ; weight with holder, 1650 grammes.

Charge: about 5500 grammes calcined alumina (Brit. Al. Co.), mixed with 1130 grammes petroleum
coke (60-mesh).

Initial pressure, 20 atmospheres coal gas.

Gaseous products of reaction washed out with coal gas (see p. 442), the gas being let off about every
10 minutes until the pressure had fallen to 15 atmospheres, a fresh supply of coal gas being at once forced
in, raising the pressure to 25 or 30 atmospheres.

TABLE IX. (2).

Time.

E.M.F. on
dynamo
terminals.

E.M.F. on
furnace.

Current.

Power.

Observations.

minutes

volts

VoUs

amperes

kilowatts

start

60

42

200

8-4

Initial pressure, 20 atmospheres coal

CfSt.fl

3

50

30

300

9-0

gas.

5

47

30

340

10-2

8

80

70

150

10-5

Gas blown off down to 15 atmospheres,

coal gas admitted to 25 atmospheres.

14

80

70

150

10-5

2nd lot of gas let off as before.

16

., 65

57

200

11-4

20

62

45

250

11-3

3rd lot of gas let off as before.

23

56

42

250

10-5

26

55

42

250

10-5

4th lot of gas let off.

31

60

48

220

10-6

34

—

5th lot of gas let off.

41

60

50

180

9-0

43

—

—

—

—

6th lot of gas let off.

45

60

50

180

9-0

56

60

50

180

9-0

stop

—

—

—

—

TABLE IX. (3).

The total volume of coal gas used during the experiment was 800 litres, and about the same volume
during the subsequent cooling period.

Product : central fused lump surrounding the vertical electrode, to which it adhered ; had a weight of
about 1560 grammes, and had the appearance of fused alumina containing some aluminium carbide.

The fused lump contained 228 grammes of aluminium carbide and 36 grammes of aluminium, some
of which was found in the form of plates after crushing.

The outside fritted material, about 1125 grammes in weight, contained a further 9 grammes of aluminium
in large pieces. No reduced material was found in the outer layers of the charge.

Total yield was therefore 228 grammes aluminium carbide 45 • 7 grammes aluminium.

rnt\.\n-: KKUTIONS

un;ii <;.\si:ors

461

TABLE X. — Experiments on the Reduction of Alumina by Carbon at Atmospheric

Pressure.

A. Radiation Heating from Arc.

Experiment

^0.

K.M.F. on
furnace.

Current.

Duration.

Product.

Observations.

G45

rolu
40

amp^rea
140

minute*
5

Fused A120S.
No Al metal
or carbide

Mixture corresponding to A1...0» + 2C
placed under arc on alumina bed.
Furnace enclosed. Hydrogen cir-
culation.

G46

25

240

14

Fused lump of
A1.0,. No
Al metal or
carbide

Similar experiment to G 45.

G47

40

300

10

No aluminium
or carbide

CaF-j used as flux (28 per cent.) with
same mixture as above. Heated
until very rapid vaporisation of
fluor spar. Furnace, ordinary
Moissan type. Material in carbon
crucible.

G48

42

300

10

No aluminium
or carbide

Larger percentage of fluoride (61 per
cent.), otherwise similar experi-
ment to G 47.

B. Resistance Core of Mixture.

Experiment
Xo.

Product

Observations.

G61

G62

Total energy, about

1 kilowatt hour
Power, 4 kilowatts

Total energy, 5'8 kilo-
watt hours
Power, 7 kilowatts

Fond lump of
alumina. No
Al or carbide

Condensate,
but negligible
Al in fused
lump

Cross-section of furnace hearth, 130sq. ecu tiros. ;
length, 20 centims. Charge, consisting of
central core of alumina and carbon sur-
rounded by pure alumina. Thick iron plates
served to cover furnace. No aluminium
condensate obtained.

Similar construction to above, but cover fitting
air-tight, and provided with long condensing
chamber. Escaping gas deposited sublimate
containing finely divided aluminium.

4G2 ELECTRIC FURNACE REACTIONS UNDER HIGH GASEOUS PRESSURES.

TABLE X. — Experiments on the Reduction of Alumina by Carbon at Atmospheric

Pressure (continued).

C. Approximate Estimation of Temperature of Reduction.

Experiment
No.

Mixture.

Temperature
estimated.

Result.

Observations.

G55

A. 10 grammes (A1203 + 2C)
+ 40 grammes Cu

M.P. Ni . . .

No appreciable
reduction

Experiment in carbon tul>e
furnace with careful ad-
justment of temperature.

B. 10 grammes (A1203 + 2C)
+ 40 grammes Cu

M.P. Pt. . .

No appreciable
reduction

Experiment in carbon tube
furnace with careful ad-
justment of temperature.

C. 10 grammes ( A120S + 2C)
+ 40 grammes Cu

Above M.P. Pt,
but below
M.P. Alj,0s

No appreciable
reduction

Experiment in carbon tube
furnace with careful ad-
justment of temperature.

G57

A. 8 grammes (A1203+2C)
+ 25 grammes Cu+10
grammes CaF2

Above M.P. Pt,
but below
M.P. A1203

No .appreciable
reduction

Experiment in carbon tube
furnace with careful ad-
justment of temperature
(mixture well fused).

B. 72 grammes (A12O3 + 2C)
+ 25 grammes Cu + 8'5
grammes CaO

Above M.P. Pt,
but below
M.P. A1208

No appreciable
reduction

Experiment in carbon tube
furnace with careful ad-
justment of temperature
(mixture well fused).

G58

A. 10 grammes (ALA + 2C)
+ 50 grammes Cu

Just above M.P.
of alumina

46 grammes
aluminium-
bronze (6 • 6
per cent. Al)

Crucible containing mix-
ture embedded in granu-
lar carbon, covered with
a second crucible con-
taining alumina. Latter
showed no sign of fusion.

B. 10 grammes ( A120S + 2C)
+ 50 grammes Cu

Considerably
higher tem-
perature

49 grammes
aluminium-
bronze (7 • 8
per cent. Al)

Similar construction
alumina in upper cruci-
ble also fused. All the
mixture either combined
or volatilised.

XII. A New Ctirri'iit H'eif/ht-r and a Determination of the Electromotive Force

of the Normal Wexton Cadmium

Hi/ /',•<,/, .,-.«>/• W. E. AYRTON, F.K.S., and'L MATHER, F.R.S., Central Technical

Coller/<\ famitnn; and F. E. SMITH, A.R.C.S., National Physical

Laboratory, Tcddington.

Received June 5, — Read June 27, IU07.

•

[PLATES 7-8.]

PRINCIPAL CONTENTS.

1'age

lintorical notes on the absolute measurement of current 464

Introductory 467

(General description of current weigher 469

Adjustable support for balance 471

The physical balance 472

Magnetic tests 475

Construction, measurement and insulation of coils 478

Axial lengths of coils 483

Diameters of coils 486

Insulation of coils 496

Erecting and adjusting the instrument 499

Advantages of duplicating the coils 505

Force between helical current and coaxial circular current sheet 507

Calculation of mutual induction of helix and circular end of coaxial current sheet 510

Differential t'ffects of the several windings and their relation to the linear dimensions of the

coils 615

Use of balance and determination of E.M.F. of cell 517

Preliminary difficulties 523

< icneral behaviour of the balance 527

Tables of results and discussion of same 529

History of the standard cell employed 535

Conclusions 538

Appendix A. Coefficients for calculation of the complete elliptic integrals F and E 540

Appendix B. On the forces between coils of wire of finite section 54 1

VOL. CCVII. — A 424. 31.1.08

464 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. K. SMITH :

HISTORICAL NOTES ON THE ABSOLUTE MEASUREMENT OF CURRENT.

A CURRENT can be measured absolutely in the electromagnetic system of units either by moans of the
action of the current on a magnet, or of the current on a current. The former method has the
disadvantage that at least two independent measurements are necessary. For example, in using an
electro-magnetic balance, the strength of the magnet acted on by the electric circuit has to be determined,
as well as the force exerted on the magnet by the circuit. In galvanometers, either of the sine or tangent
type, the magnetic field produced by the electric circuit is compared with the earth's horizontal field, the
strength of which is determined independently. Further, as the strength of artificial magnets cannot be
regarded as truly constant, and the earth's field is subject to diurnal and secular variations, this class of
measurement is not ideal.

In the electrodynamic class of measurement the mutual action between two or more coils carrying
current takes the form of a torque, as in electrodynamometers, or a direct force, as in current weighers.
In electrodynamometers the torque may be measured with a bifilar suspension, the torsion of a wire or
spring, or by means of a gravity balance. Current weigher measurements are almost always made by
direct comparison with gravity, which is believed to be constant, and is known to a higher degree of
accuracy than the strengths of any magnet or magnetic field that has yet been measured.

Shortly after the absolute system of units was devised by GAUSS and WEBER in 1832, A. BKCQUEUEL*
weighed the attraction between a coil and a magnet ; and two years later LENZ and JACOBI t used and
modified BECQUEREL'S balance by arranging a coil and magnet at each end of the beam. In 1840
W. WEBER determined the electrochemical equivalent of water, using the tangent galvanometer as his
instrument for measuring current; and in 1843 similar measurements were made by BUNSEN and by
CASSELMANN, followed in 1851 by JOULE.

Meanwhile W. WEBER| had, in 1846, invented his two forms of electrodynamometer, one with the
suspended coil inside, and the other with this coil outside the fixed coil, and he measured the torque with
bifilar and unifilar suspensions.

The first current weigher appears to have been constructed by CAZIN§ in 1863. This consisted of two
rectangular coils with their planes horizontal, one hanging from the beam of a balance directly alx>ve the
other, which was supported on an adjustable table. The instrument was used for determining the electro-
chemical equivalent of water.

In 1864 JOULE |i made a current weigher having three circular flat coils wound with copper strip, one
being suspended from a balance, so that its mean plane, which was horizontal, was midway between those
of the other two fixed coils. This instrument had the correction to its principal constant determined by
comparison with a standard tangent galvanometer, and was employed in JOULE'S electrical determination
of the mechanical equivalent of heat. Its object was to enable a constant current to be maintained through
the calorimeter, independent of variations in the earth's magnetic field.

LATIMER CLARK,H in determining the E.M.F. of his standard cell in 1872, used a bifilar olectro-
dynamometer with circular fixed and moving coils, each arranged in the Helmholtz fashion. The fixed
coils were of large size relative to the suspended ones, a fact which considerably simplified the calculation
of the torque per unit current. The instrument had been constructed for the Electrical Standards

* 'Comp. Rend.,' vol. V., p. 35, 1837.

t ' POGG. Ann.,' XL VII., p. 227, 1839.

J 'Electrody. Mess.,' Vol. I., p. 16, 1846.

§ ' Ann. de Chim.,' [4], Vol. I., p. 257.

|| ' B.A. Report,' 1864.

f « Roy. Soc. Proc.,' May 30, 1872 ; also ' Phil. Trans.,' 1874, Part I.

A NEW CURRENT WEIGHER, ETC. 465

Committee of the British Association, and was wound by CLERK MAXWK'I.I.. LATIMKR CI.ARK also used
a sine galvanometer for his E.M.F. measurements, and arrived at the values 1-4573 and 1 -4562 B.A. voltt
at 15-5'C. with the two methods respectively.

In 1873 F. KOHI.RAUHCH* employed the tangent galvanometer and magnetometer in determining the
electrochemical equivalent of silver, which he found to be 1 • 1363 milligrammes per coulomb.

MAscART.t in 1882, constructed his current weigher formed of a long solenoid hung from a balance
arm, with its lower end in the mean plane of a large circular coil, and published the number 1*124 milli-
grammes as the mass of silver deposited by one coulomb. This was corrected in 1884 to 1 • 1 15G.J

At the British Association Meeting in 1882§ Lord RAYI.KIGH discussed the several methods of
measuring current absolutely which had IK-CM employed by previous experimenters, more especially those
used by KOHI.RAUSCH and by MASCART. He pointed out that a large part of MASCART'S long solenoid
was comparatively ineffective, and considered that the moving coil should be compact and situated near
the position of maximum effect. A further advantage would, he pointed out, be gained by duplicating the
fixed coil, thus making the arrangement symmetrical and doubling the force. The dimensions of (current)*
in the electromagnetic system lieing the same as those of force, Lord KAYLKICH showed that the constant
of a current weigher arranged as described above, must be a numeric, depending on the mean radii of the
coils as a ratio, which could be determined electrically with high precision without any linear measurements
whatever having to be made.

In 1883 Lord RAYI.KIGH published the result that he had obtained with a current weigher thus
constructed, viz., 1-119 milligrammes of silver per coulomb. Meanwhile F. and W. KOHI.RAUSCH had
carried out measurements of high precision with the tangent-galvanometer and suspended-coil method,
obtaining the values 1-11833 and 1-11822 respectively in 1881 and 1883, although these results were
not published until later.f

In a classical memoir** Lord RAYI.KIGH and Mrs. SIDGWICK showed that the number given by Lord
RAYI.KICH in 1883, viz., 1-119, was too high by nearly 1 in 1000, owing to inclusion of mother liquor
with the silver. This was due chiefly to the solution being filtered through silver acetate to secure firmer
deposits. With pure silver nitrate they found the equivalent to be 1 -11794, the greatest difference from
the mean of thirteen experiments being less than 1 part in 2500. The paper contains a full description
of the current weigher, the method of using it, the calculation of the force between the coils, and
a table of numbers for facilitating the making of these calculations by elliptic integrals. Also a very
careful determination of the E.M.F. of a number of CLARK cells is given. It is important to notice that
no measurements of length, moment of inertia, or time are necessary in determining current with a
current weigher made on Lord RAYLEIUH'S plan, and this constitutes one of its great advantages.

THOMAS GRAY, in 1886,tt determined the electrochemical equivalent of silver by means of a sine
galvanometer of his own design, and in 1887 KoKPSELjt used an electromagnetic balance of most ingenious
construction, made according to VON HKLMHOLTZ'S instructions, for the same purpose. The results
obtained, although approximating closely to those of F. and W. KOHI.RAUSCH, and of Lord RAYLKIOH and
Mrs. SIDGWICK respectively, are not so trustworthy.

* 'Pooo. Ann.,' 149, S. 170, 1873.

t 'Jour, de Phys.,' [2], t I., p. 109, 1882.

t ' Jour, de Phys.,' t. III., p. 283, 1884.

§ 'B.A. Report,' p. 445, 1882.

|| ' Proc. Cambridge Philosophical Society,1 vol. V., p. 50.

IT • Site, der Phys.-Med. Ges. ru Wiirzburg,' 1884 ; also « WIKD. Ann.,' 27, p. 1, 1886.
** 'Phil. Trans.,' 175, p. 411, 1884.
ft ' 1'hil. Mag.,' 22, p. 339, 1886.
U ' WIED. Ann.,' 31, p. 250, 1887.
VOL. CCVII. A. 3 O

466 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

In 1890 PELLAT and POTIER* employed an electrodynamometer balance in silver deposit work, which
had a short cylindrical coil secured with its axis vertical to one arm of a balance ; this arm projected along
the axis of a long horizontal solenoid fixed symmetrically with respect to the moving coil. The torque
between the coils was balanced by weights, the magnitude of which gave 1-1192 as the mass of silver
deposited per coulomb.

With a view to simplifying the use of RAYLEIGH'S current weigher, HEYDWEiLi.ERt in 1891 modified
the arrangement by placing the coils with their common axis horizontal, the moving coil being carried
directly below the centre of the balance beam. Nearly the whole of the force was balanced by weights
on the horizontal arm, and the rest determined from the slight displacement of the coil from the vertical
position.

To determine the E.M.F. of CLARK cells in 1896 KAHLE| used a HELMHOLTZ electrodynamometer
balance of novel construction, in which the moving coil and balance beam were supported by, and so that
they rolled on, thin metal strips which served also as leads. Rectangular coils of many turns embraced the
balance case in planes perpendicular to the length of the beam. The constants of these coils, as well as of
the suspended one, were determined by comparison of their magnetic effect with that of a large rectangle
of copper band stretched round a strong metal frame, the dimensions of which could be accurately
measured. The experiments gave the result 1-4322 at 15°C.

In 1897 the late Professor J. VIRIAMU JONES, in collaboration with one of the authors (W. E. A.),
devised a current weigher in which the forces could be calculated with great exactness by a formula
developed by the former,§ and a preliminary instrument was constructed with single layers of wire in
screw grooves, and described at the British Association Meeting in 1898.||

Messrs. PATTERSON and GuTHE,5I working under Professor CARHART, employed a torsion electro-
dynamometer with fixed coils on wood and suspended coil on vulcanite, and made determinations of silver
deposit (1-1192 milligrammes per coulomb) which they believed accurate to 1 part in 5000. In the
following year (1889) CARHART and GUTHE** measured the E.M.F. of CLARK cells with the same
instrument, obtaining the value 1-4333 at 15° C., and in 1902 CALLENDARfT published the result (1-4334
at 15° C.) got by R. 0. KING with an electrodynamometer of the British Association pattern employed in
his (CALLENDAR'S) researches on " Continuous Electric Calorimetry."

Further determinations of the electrochemical equivalent of silver with PELLAT'S electrodynamometer
balance were made in 1903 by PELLAT and LEDUC,JJ who obtained 1-1195 milligrammes per coulomb. In
the same year VAN DlJK and KUNST§§ carried out a very careful research in a new laboratory free from
iron and vibration, using two tangent galvanometers, magnetometer and variometer, and from the mean of
twenty-four closely accordant determinations of the electrochemical equivalent of silver deduced the value
1 -11818. This they believed to be accurate to 1 part in 10,000.

Professors CARHART and PATTERSON || || described, at the meeting of the Electrical Congress at St. Louis

* 'Jour, de Phys.,' t. VI., p. 175, and t. IX., p. 381, 1890.

t ' WIED. Ann.,' 44, p. 533, 1891.

J 'WlED. Ann.,' 59, p. 532, 1896.

8 'Roy. Soc. Proc.,' vol. 63, p. 204, 1898.

|| ' B.A. Report,' Bristol, 1898, p. 157 ; also ' Jour. Inst. Elec. Eng.,' vol. 35, p. 12, 1905.

f 'Phys. Rev.,' VII., p. 257, 1898.
** 'Phys. Rev.,' IX., p. 288, 1899.
tt 'Phil. Trans.,' A., 199, p. 81, 1902.
it 'Comp. Rend.,' 136, p. 1649, 1903.

§ § ' Verel. van de Gewone Vergadering der Wis- en Natuurkundige Afdeeling,' Dec., 1903.
III! 'Jour. Inst. Elec. Eng.,' vol. 34, p. 185, 1905.

A NEW CURRENT WEIGHER, ETC. 467

in 1904, a now torsion electrodynamometer of the GRAY* pattern, having single-layer coils on cylinders
i if Paris plaster. Experiments on CLAKK and cadmium cells were then in progress.

Last year (1906) GuTHEf published the results of a lengthy research on CI.AKK and cadmium cells in
which another GRAY electrodynamometer was employed. He arrived at the values 1*43296 at 15" C. and
1 -01853 at 20° C. for the respective cells, and deduced from this and previous work 1 '11773 milligrammes
per coulomb as the electrochemical equivalent of silver. The instrument employed by GITHE suffers from
non-uniformity of winding, but this was allowed for approximately. Its influence on the accuracy of the
electrodynamometer is discussed by ROSA in the same number of the ' Bulletin ' (p. 71).

SECTION 1. — INTRODUCTORY.

The instrument herein described is the outcome of conversations between the late

I'lnt'rssnr -I. YllMAMU .li'M-ls ami MM.- of (lie :tlltliol> I W. K. A.) '•!! tllril' ivtlini tVoln

the British Association meeting, held in Toronto in 1897.

Absolute determinations of resistance had been made on many occasions, and with
considerable precision, whilst those of current were comparatively few ; the want of
agreement between the results obtained by different observers was by no means
satisfactory. It was therefore decided to make a new determination of the ampere by
means of a current weigher formed of coils with single layers of wire, such as had been
so successfully employed by Professor JONES in his determination of the " Specific
Resistance of Mercury in Absolute Measure" ('Phil. Trans.,' A, 1891), and by
Professors AYRTON and JONES in their determination of the ohm at the Central
Technical College, London, in 1897.J

By using coaxial coils, with single layers of wire wound in screw-thread grooves,
advantage could be taken of the convenient formula developed by Professor JONES for
calculating the electro-magnetic force between a helix and a circular current sheet,§
viz.,

where yk is the current in the helix, y the current per unit length of the current
sheet, and M,, Ma the coefficients of mutual induction of the helix and the two circular
ends of the current sheet.

To test the stability of the proposed current weigher, or " ampere balance "
as it is frequently called, as well as to get experience regarding the conditions
necessary for successful operation, a preliminary apparatus was constructed at
the Central Technical College in 1898, and there used to make an approximate

* ' GRAY'S Absolute Measurements, &c.,' vol. 2, part 1, p. 274.

t ' United States Bureau of Standards Bulletin,' vol. 2, No. 1, p. 33, 1906.

t 'B.A. Report,' Toronto, p. 212, 1897.

§ " On the Calculation of the Coefficient of Mutual Induction of a Circle and a Co-axial Helix, and of
the Electromagnetic Force between a Helical Current and a Uniform Co-axial Circular Cylindrical Current
Sheet," 'Roy. Soc. Proc.,' vol. 63, p. 204, 1898.

3 o 2

468 PKOFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

determination of the electro-chemical equivalent of silver.* In this instrument
the coils were formed by winding insulated wire in the grooves of screw threads
cut in metal cylinders, but the springiness of the covering prevented very exact
measurements of the dimensions being made. To obtain greater precision, it was
decided to use, in the proposed balance, bare wire wound on insulating material,
as originally employed in the Lorenz apparatus designed by Professor J. V. JOM.S
for the McGill University, Montreal, and to avoid the uncertainty as to leakage
between adjacent turns of such a spiral t the arrangement devised by one of the
authors (W. E. A.) of having double-threaded screw grooves wound with separate
bare wires, subsequently connected in series after the insulation resistance between
them had been made satisfactory, was adopted.

Experience with the preliminary apparatus showed that air convection currents
should be minimised, and that easy access to, and independent adjustments of, both
fixed and suspended coils were very desirable. In designing the new current weigher,
in collaboration with the late Professor JONES, these points were kept in view, and
the arrangements chosen were such as would take full advantage of the mechanical
precision attainable with modern machine tools, a subject which Professor JONES had
very much at heart. In fact, he had long advocated that the instruments employed
in realising the concrete values of the electrical units from their absolute definitions
should be engineering tools rather than ordinary physical laboratory apparatus.

Complete working drawings and specifications of the proposed instrument, and its
adjustable support, were prepared at the Central Technical College during the Session
1898-99, the drawings being made by Mr. J. P. GREGOEY, then a student of the
College, and now of the British Thomson Houston Co., Rugby. Tenders were
obtained for the construction of the instrument, to defray the cost of which the
British Association for the Advancement of Science made a grant of £3004

As the amounts of the tenders for the balance, and the adjustable phosphor-bronze
stand for supporting it, much exceeded the above-named sum, Sir ANDREW NOBLE,
F.R.S., was approached, and took so much interest in the apparatus and the important
work that was to be carried out with it, that he generously presented the carefully
made adjustable support, constructed by Messrs. SIR W. ARMSTRONG, WHITWORTH
and Co., Limited, free of cost.

The physical balance was built by Mr. L. OERTLING, of London, and the electrical
portions were made at the National Physical Laboratory, under the supervision of the
Director, Dr. R. T. GLAZEBROOK, F.R.S.

We may here remark that the current weigher has proved to be the most perfect
absolute electrical instrument hitherto constructed, and has enabled us to determine

* 'B.A. Report,' Bristol, 1898, p. 157; also 'Jour. Inst. Elec. Engrs.,' vol. 35, p. 12, 1905.
t This uncertainty necessitated the removal of the original winding of the Lorenz apparatus, and
rewinding with silk-covered wire. See ' Jour. Inst. Elec. Engrs.,' vol. 35, p. 1 3.
J 'B.A. Report,' 1898, p. 147.

A NKW CURRENT WEIGHER, ETC.

409

the ampere to a very high degree of accuracy. In fact, this unit is now known with
a precision considerably greater than any other electrical quantity of which absolute
measurements have been made.

SECTION 2. — GENERAL DESCRIPTION.

The instrument consists of a very sensitive physical balance supporting a coil with
vertical axis from each end of the beam, these coils hanging coaxially within fixed
coils carried from the base of the balance. A diagrammatic sketch of the arrangement
is shown in fig. 1, and a view of the complete instrument in fig. 2, Plate 7.

From the former it will be seen that the current flows in opposite directions in
the upper and lower parts of the outer coils. On the left-hand side of fig. 1 the
current in the upper half of the outer coil flows clockwise (looking from above) and in
the lower half counter-clockwise, whilst in the left-hand suspended coil the circulation
is shown clockwise. The tendency is, therefore, to lift the suspended coil SL. It

Fig. 1. Diagram of windings.

Fig. 3. Diagram showing hollow
cylinder with double winding
in grooves of screw threads.

will also be seen that the outer coils on the right will tend to depress the suspended
coil SR, so that the two sets of coils exert a clockwise torque on the beam. This
torque is balanced by weights added to or taken from scale pans supported
independently on the knife edges which carry the suspended coils, an arrangement
which avoids displacement of the suspended coils when the weights are placed or
removed.

All the coils are wound with bare wire on hollow marble cylinders, having double-
threaded screw grooves cut on the surfaces, into which separate wires are laid as
shown in fig. 3. In this figure one wire is indicated by two thin lines, and the other
is shown thick. The two wires, hereafter distinguished as No. 1 and No. 2, form two
adjacent helices, which, in the use of the instrument, are connected in series and act
as one coil. They can, however, be readily disconnected from each other and an

470

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

insulation test made between them. This applies to each of the six coils forming the
current weigher, arrangements being made whereby the six No. 1 wires may be
connected together, the six No. 2 wires similarly grouped, and the insulation between
adjacent wires of the whole instrument tested simultaneously. Any leakage between
the two adjacent helices can thus be readily detected and localised and remedied.

Each of the fixed cylinders carries four helices, two upper and two lower, and each
suspended cylinder two. There are therefore twelve helices in all, and these are connected

Fig. 4. General view of instrument, showing outer coils lowered.

in series in the normal use of the current weigher by means of small concentric cables
running to a plug board and commutators outside the balance case. Flexible con-
nections are used as leads and returns to and from the suspended coils. The
commutators enable the direction of the current in any coil to be changed at will. By
reversing the current in the coils on the fixed cylinders the forces between the fixed
and suspended coils are reversed, and the apparent change of weight thus produced
is a measure of the square of the current used.

A NEW CURRENT WEIGHER, ETC.

471

The position of the balance beam is observed by viewing a finely divided scale
carried by the pointer through a microscope seen in fig. 2, Plate 7, and in fig. 4.

A double glazed case or cover, with f-inch air space between the sheets, resting
on a phosphor-bronze plate, serves to exclude dust and draughts, and to minimise
convection currents which may be caused by unequal radiation or conduction from
surrounding objects.

The whole instrument is supported on an adjustable phosphor-bronze stand or
pedestal at a convenient height (see fig. 4), levelling screws being placed at the
corners of the base.

SECTION 3.— ADJUSTABLE SUPPORT FOB BALANCE.

On opposite sides of the central pillar of the pedestal (see fig. 4) are sliding
brackets BB, like the tables of small milling machines, which can be lowered through
distances of about 14 inches (35 centims.) by means of vertical screws SS. Each
bracket supports a slide rest having a circular top-plate which can be moved
half-an-inch horizontally in two directions at right angles by means of screws with
graduated heads. The nuts on the vertical screws are of large diameter, and they

Fig. 5. Section through top-plate of slide rest for supporting fixed cylinders.

and the heads of the horizontal screws are divided to read thousandths of an inch.
As each division can be subdivided by eye to tenths, it is possible to read the position
of either fixed cylinder to a ten-thousandth of an inch.

The weights of the fixed cylinders and brackets are sufficient to overcome the
friction in the vertical slides and thus avoid backlash in these motions. In the
horizontal movements backlash is avoided by using strong phosphor-bronze springs
shown at a, fig. 5, capable of moving the corresponding slide when tightened up

472 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

to prevent shake and loaded with a fixed cylinder. These springs keep the
horizontal screws always in tension.

When the brackets B, figs. 4 and 5, are near their highest positions, the circular
top-plates P'P' of the slide rests project through holes in the phosphor-bronze plate,
PP, fig. 5, which forms the base of the balance. Copper spinnings s of section ~\ fit
closely round the top-plates, and can slide between the plate P and ring r, thus
forming a draught-tight joint, and at the same time permitting horizontal motion of
about half-an-inch in any direction.

For supporting the marble cylinders M, fig. 5, annular phosphor-bronze castings C,
of inverted channel section, rest on fine-threaded levelling screws I, projecting through
the top-plates of the slide rests, the heads being below the plates, so that levelling
can be done from beneath the balance case. This arrangement is on the " hole, slot
and plane principle," to avoid constraint and yet ensure precision in position.

SECTION 4. — THE PHYSICAL BALANCE.

A photograph of the instrument, without coils, is shown in fig. 6, Plate 8. It
has a beam 20 inches (50'8 centims.) long, capable of supporting 5 kilogrammes at
each end, and turning with about one-tenth of a milligramme, a rider beam, divided
into 100 parts on each side, and two rider carriers are fitted. All the knife edges
and planes are of agate, and as fine as possible consistent with the loads they have
to carry.

From each of the outer knife edges K there depends a three-armed spider S, with
heavy nuts N at the end of each arm, and adjustable hooks a, a, from which the
corresponding suspended cylinder hangs on three phosphor-bronze wires w, w, w.
The object of the nuts is to enable the suspended cylinder to be levelled, two very
sensitive levels being fixed to the cylinder for this purpose.

Below the suspended cylinder, and quite clear of it, is a copper disc d, fig. 6,
carried by three wires w', w', w' attached to the clamping beam F of the balance, for
supporting the cylinder should one of the wires w, w, w become unhooked.

The scale pans for carrying the weights used to balance the forces exerted by the
coils, hang from separate planes on the same knife edges as support the suspended
cylinders. These may be seen in fig. 7, where K is the knife edge, H is the hook
carrying the spider S, and h the hook supporting the scale pan p. This arrangement
is novel, and of considerable utility, for it permits of removal or replacement of the
weights without affecting the levelling of the suspended cylinder. Its adoption,
however, necessitates the perfect straightness of the knife edges. This condition has
been satisfied to a very high degree of accuracy by Mr. OERTLING, for shifting a
weight of 16 grammes from the scale pan to the cylinder produced no appreciable
difference in the rest-point of the balance, when the sensitiveness was such that one-
tenth of a milligramme could be detected.

As will be seen from figs. 2, 6, and 7, the scale pans are of unusual shape. Rods

A NEW CURRENT WEIGHER, ETC.

473

<%nd Weight Lifters.

Fig. 7. End and side elevations to show mode
of supporting scale pan and spider from
same knife edge.

VOL. OCVII. — A.

Fig. 8. Arrangement of leads to coila on fixed
cylinders.

3 P

474 PROFESSOR W. E. AYKTON, MR. T. MATHER AND MR. F. E. SMITH :

R, R, R, fig. 7, project upwards from the plate p', and from the middle and upper ends
of these rods sector-shaped pieces q project inwards and form tripods on which the
weights may rest. Claw-shaped lifters on the arms A, A, figs. 6 and 7, are operated by
cams C fixed in the corners of the balance case, and serve to remove or replace the
weights. The arrangement is very convenient, and works with perfect smoothness,
the result of the excellent workmanship of Mr. OERTLING. Two weights and two
lifters are provided at each end of the balance. These may be seen in the end view
of the instrument shown in fig. 7a, Plate 8, and also in fig. 6.

Another novel feature of the balance is the arrangement employed for taking the
beam off the centre agate plane and fixing it in the zero position without appreciably
raising or lowering the suspended cylinders. This is of considerable importance, as it
allows of the coils being levelled and adjusted vertically to the sighted position,
without continually clamping and freeing of the beam for making and testing the
adjustment. The specification for the balance stated that " the displacement of the
suspended coils caused by fixing the beam must not exceed 4 mils (one-tenth of a
millimetre)." Mr. OERTLING has, however, used a construction which reduces the
displacement to a far lower figure, as the fixing is effected without raising the beam
more than 0'004 millim. (4 microns), and the planes carrying the suspended coils are
clamped with a movement less than 0'08 millim.

The handle seen at the front of the case in figs. 2 and 6 actuates the clamping
arrangements. Turning it clockwise through 180° from the position shown lowers
the whole clamping beam F, fig. 6, thus bringing the centre knife edge against its
plane, and allowing the planes supporting the scale pans and cylinders to rest on the
end knife edges. By sliding inwards a tube surrounding, the horizontal clamping
axle, and turning the handle through another 90°, two agate hemispheres m, m are
brought into contact with agate planes on the beam immediately above them and fix
the beam in the zero position whilst the scale pans and cylinders still hang on the
beam. This device is made use of when changing the weights, and on reversal of
current in making measurements.

For observing the rest-point of the balance a microscope, seen at Ma, fig. 6, is used
to view a finely divided ivory scale carried by the pointer at a distance of 14f inches
(37 '2 centims.) from the knife edge. The magnifying power of the microscope is
about 48, and the scale is f of an inch long, divided into 200 parts ; each division is
therefore ^§0'' (0'095 millim.). The cross wires and the lines on the scale are
sufficiently fine to permit of one-twentieth of a division to be estimated quite easily,
and with care and practice it is possible to read to fiftieths of a division, and in some
cases to hundredths.* In all observations the method of vibrations was employed in
determining the rest-point, the amplitude being limited to a few divisions on either
side of the middle.

* For illuminating the scale, a lens and a Nernst lamp placed some 6 feet away were used, and proved
most satisfactory.

A NEW CURRENT WEIGHER, ETC. 4/5

To allow of free access to the balance there are two sliding sashes, S', S', fig. 6, both
at the front and back of the case, and the ends have hinged doors, D, opening out-
wards. The middle portion of the case carrying the microscope and the corresponding
piece at the back can also be removed. It is thus possible to make any adjustment
required with comparative ease. Fig. 2, Plate 7, shows a view of the balance with
sides of the case taken away.

Although it is not essential that the arms of the balance should be of equal length
in a current weigher used in the manner described on p. 526, it was thought desirable
to determine the ratio of their lengths. Employing weights of 50 grammes, it was
found that

length of left arm -5- length of right arm = 1 "00001,,

so a very close approximation to equality exists.

SECTION 5. — MAGNETIC TESTS.

As it is of considerable importance that the permeability of all parts of the current
weigher be practically unity, magnetic tests were made on the materials employed.

Before the phosphor-bronze support for the balance was cast, Sir ANDREW NOBLE
forwarded to the Central Technical College in September, 1899, a bar of the alloy it
was intended to use, and careful experiments were made on the material. Tested by
a very sensitive magnetometer the bar showed no magnetic property. An induction
balance having two primary coils in series, and two secondaries in series opposing, with
a sensitive moving-coil galvanometer in the same circuit, was therefore set up. One ot
the induction coils was in the form of a solenoid 2'4 centims. diameter and 36 centims.
long, wound with 457 turns of No. 18 S.W.G. wire as primary and 1600 turns of
No. 34 S.W.G. as secondary. The other half of the balance was formed of two
separate coils whose relative position could be varied continuously until their mutual
induction exactly balanced that of the solenoid windings when the core was of air.
By shunting a known fraction (joW) °f the current from the primary of the second
pair the swing obtained on the scale of the galvanometer gave a measure of the
sensitiveness of the arrangement ; this was sufficient to show a change of 1 part in
30,000.

On removing the above-mentioned shunt and inserting the phosphor-bronze rod
(2 centims. diameter and 30 centims. long) into the solenoid, a quick jerk of the
galvanometer spot was olwerved on starting and stopping the primary current, and a
rapid return to zero. The direction of the kick was such as would be produced by a
permeability less than unity ; the effect, however, was traced to be mainly due to
eddy currents in the rod, and was nearly neutralized by putting a tertiary coil, with a
resistance box in series with it, in proximity to the second pair of coils. The
resistance in the tertiary circuit could be so adjusted that the movement of the spot

3 P 2

476 PROFESSOR W. E. AYRTON, MR. T. MATHER AJsD MR. P. E. SMITH:

on starting or stopping the current was barely perceptible when the bronze rod was
inside the solenoid, and on removing the rod and opening the tertiary circuit, without
making any other change whatever, the balance was to all appearances perfect. We
were therefore certain that the permeability of the alloy differed very little, if at all,
from unity, so the casting of the stand was proceeded with.

Similar tests were made on the completed stand when received at the Central
Technical College in 1900. The shape and size of the stand, however, made it
difficult to place within coils of manageable dimensions, so a modified method of
testing was used. The College possessed a standard of mutual induction (called S
in this section), of O'Ol henry, made in 1892, consisting of coils wound in grooves on a
wooden disc 9£ inches diameter and 2f inches thick, so it was decided to test the
stand by observing whether the mutual induction of these coils was altered by
placing them on the circular top-plates of the slide rests which were to support the
coils. To do this, an induction balance formed of the mutual induction standard S
and another pair of coils was arranged as described above. The system was carefully
balanced when S was supported on one end of a pine* board, 1 inch by 11 inches by
12 feet long, the other end of which rested on one of the top-plates. On moving S
to the middle of the board the balance was not disturbed, but on placing it over the
stand a quick jerk of 85 divisions and rapid return to zero was noticed. This kick,
the effect of eddy currents in the metal of the support, was neutralised as far as
possible by a tertiary circuit. It could not, however, be entirely eliminated by the
tertiary coils available, a phenomenon attributed to want of equality in the time-
constants of the tertiary circuit and of the eddy-current circuits in the continuous
metal. The procedure adopted was to observe the swing produced by shunting 7^0
of the current from the primary of the balancing pair of coils, when the test pair were
supported above air, and on the top-plate of the stand respectively, the tertiary
circuit being open in the former case and closed in the latter. In each of the two
positions the swing produced was 33 divisions. The sensitiveness of the arrangement
was thus I in 33,000 per division, and under these conditions no difference would be
detected. Four sets of tests were made giving precisely equal swings.

The experiments were repeated on the top-plate of the second slide rest of the
stand with the same result. The eddy-current effect was somewhat different in the
two cases, for in one the resistance in the tertiary circuit necessary to give minimum
kick was 134 ohms and in the other 124 ohms.

To test whether the two ends of the pine board differed magnetically, it was
turned end for end, and the whole cycle of operations repeated. No difference could
be detected. In all cases great care was taken to twist the leads together in pairs, so
as to avoid mutual induction in parts of the circuit other than that under test. The
test coils S and the balancing coils were kept far apart and with their planes at right

* Pine was used because previous work in connection with very sensitive moving-coil galvanometers
had shown this material to be non-magnetic.

A NEW CURRENT WKIUHKK, ETC. 477

angles, so that there was no mutual induction t>etween the members of one pair and
those of the other pair.

After the complete ampere balance was set up at the National Physical Lalwratory,
further magnetic tests were made on the stand and surrounding parts by sending*
current of 1 ampere (approximately) round one of the suspended coils only, and
observing whether the rest-point of the balance was affected thereby. The same test
was made with the current reversed, and the whole repeated on the other suspended
coil. In neither case could any change in the rest-point be detected. Experiments
were also made by bringing masses of iron in proximity to the balance when the
suspended coils were carrying their normal current. The effects of these masses were
much smaller than expected ; in fact, the iron had to be placed very near a current-
carrying coil to produce any observable change on the rest-point. It may, therefore,
be concluded that there can be no appreciable error in the t>alance due to magnetism
or diamagnetism of the phosphor-bronze support.

Magnetic tests on marble were made at the Central Technical College in 1897,
using the large marble cylinder employed in the Lorenz apparatus constructed by
Messrs. N ALDER BKOS. & Co. for the McGill University, Montreal.* Its permeability
differed from that of air by an amount too small to be detected, t This fact, together
with the high specific resistance of marble, decided the material to be used for the
cylinders of the proposed current weigher.

All the marble used in the fixed and suspended cylinders of the ampere balance
was tested at the National Physical Laboratory, when received from the merchants,
by observing the swing (if any) of a galvanometer in the secondary circuit of a pair of
coils when the marble was quickly inserted as a core, the current in the primary
circuit being kept quite constant. The primary coil had 1000 turns of No. 32 S.W.G.
copper wire, and the secondary 10,000 turns of No. 42 S.W.G. With a current of
0'5 ampere in the primary the arrangement was extremely sensitive, as a change in
the primary current of 1 part in 10,000 produced a swing of 5 '4 millims. The scale
could be read to 0'2 millim., so that a change of flux of 1 part in 270,000 could be
detected. The tests showed that the permeability of the marble did not differ from
that of air by 1 part in 100,000, a result which is in agreement with the American
measurements mentioned above.

By means of the same coils the susceptibility of solid ferrous sulphate was measured
as 73 x 10"*, crystallised salt being used, and the air space determined by the aid of
alcohol. KONIGSBERGER! gives 37 x 10~* as the susceptibility of powdered ferrous
sulphate.

* 'B.A. Report,' 1897, p. 218.

t More recently tests made in America by WILLS, GUTHK, and STKBBINS, show the magnetic
susceptibility of several kinds of marble to be extremely small, probably less than 1 x 10"'. Se« ' Bulletin
of Bureau of Standards,' vol. 2, pp. 52, 89.

I • WlED. Ann.,' 66, 698, 1898.

478 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

The castings, drawn tubes, screws, &c., intended for making the beams, pillars, and
other parts of the physical balance, were forwarded by Mr. OERTLING to the National
Physical Laboratory for magnetic test to determine their suitability, or otherwise, for
the purpose. With the exception of the castings, all the first samples showed distinct
paramagnetic properties, and were rejected, as also were several specimens submitted
subsequently. It was found necessary to obtain the tubes and rods from special
sources before satisfactory ones were secured. Many samples of brass screws were
purchased and tested, but none were sufficiently free from magnetism, so Mr. OERTLING
was obliged to cut all used in the instrument in his own shops. The locks and keys
for the balance case had also to be specially made, phosphor bronze being the chief
material used. No trouble was experienced with the paraffin wax used to coat the
windings.

In the magnetic tests on the metallic parts of the physical balance the eddy-
current effects were small. When a brass rod was quickly inserted as a core to the
primary, a swing of 4 millims. resulted, the direction being the reverse of that
corresponding to increased permeability. To overcome this effect a thick brass rod
was cut in two and the metallic substance placed between and in contact with the
two portions, the whole being connected together by means of a metal tube so as to
realise as nearly as possible a continuous metallic conductor. The brass rod was so
long that when introduced into the primary it projected about 40 centims. from the
far end when the metal under test was about to enter the coil. An axial motion of
the metallic rod did not give rise to eddy currents capable of producing a swing of
0'2 millim.

Further tests on the suspended system were made after the erection of the balance
by sending a current through one of the fixed coils when lowered so that the
corresponding suspended cylinder was without it. The rest-point of the balance was
unaffected thereby, and remained unchanged when the current was reversed. Similar
observations were made when the current was sent through the other fixed coil, but
no change was detected.

SECTION 6. — CONSTRUCTION, MEASUREMENT, AND INSULATION OF COILS.

Preliminary tests at the Central Technical College and subsequent ones at the
National Physical Laboratory led to the choice of " First Statuary " Carrara marble for
the material of the cylinders. The tests showed this to be an excellent electrical
insulator and of negligible magnetic susceptibility. The preliminary insulation tests
were made on a small cylinder 4 inches in diameter and 2 inches in axial length. A
double screw thread (36 turns to the inch) was cut on this, and helices of No. 24
bare copper wire wound thereon. The insulation resistance between adjacent strands
was low at first, but rose to 4000 megohms when the cylinder was immersed in hot
liquid paraffin wax, removed, and allowed to cool. The magnetic tests have already
been described.

A NEW CURRENT WEIGHER, ETC. 479

The cylinders were prepared in the rough by Messrs. GOODY and CRIPPS, the large
ones being 13 inches in diameter, 11 inches in axial length, and 2 inches thick. The
corresponding dimensions of the small cylinders are 8, 6, and 0'5 inches. A few
veins run through the large cylinders, but the dark material, of which these consist,
is of negligible magnetic susceptibility. An appreciable quantity of the substance
was collected from a number of rough pieces of marble sent by the marble merchants,
and this was subjected to the magnetic tests already dealt with ; there was no
indication that the permeability differed from unity.

The cylinders are of an inconvenient shape and size for a direct determination of
their coefficient of expansion ; moreover, it was inadvisable to immerse them in water,
and this latter operation was desirable if satisfactory observations were to be made. A
bar of marble, 45 x 5 x 2*3 centims., was therefore procured from the same source;
this was baked in an oven at 140° C. and soaked in hot paraffin wax previous to any
linear observations being made. The mean coefficient of expansion between 1° C.
and 25° C. was determined by Mr. ATTWKLL to be 24 x 10~7 per 1° C.

The marble cylinders were examined for flaws and freedom from cavities ; they
were then turned until their dimensions were approximately correct, and afterwards
baked in an oven at a temperature of 140° C. for 30 hours. On the completion of
the baking, and whilst in a hot condition, they were immersed in hot paraffin wax at
110° C. No bubbles of gas were evolved from either of the four cylinders used in
the ampere balance, but from one part of another cylinder, which was rejected for
reasons mentioned hereafter, a tiny stream of bubbles escaped for a minute or two
after immersion in the wax. Each cylinder remained immersed for at least 36 hours ;
on removal it was again examined for flaws, but none were detected. Previous
to the turning of the marble cylinders, a long steel rod was turned on the lathe
set apart for this work, and its ellipticity and conicality were determined by
measurement. The ellipticity was very small and the lathe was adjusted until the
conicality was too small to be measured with certainty ; notwithstanding, the
marble cylinders turned subsequently are distinctly conical, and, in the case of the
large cylinders, those ends are the larger which were nearer to the face plate when
the spiral grooves were cut. We conclude, therefore, that the weight of a cylinder
produced a tilt, and that better results might have been obtained by turning between
dead centres. The two small cylinders were turned in this way.

A cylinder was secured to the face plate of the lathe by four external dogs, the
space between the face plate and the end of the cylinder nearest to it being about
l£ inches. This mode of support enabled the two ends to be turned truly parallel,
and the interior surface to be turned normal to them. To turn the outer surface,
four large metal studs were turned in position on the face plate, and one end of the
cylinder fitted over them ; this end was pressed into contact with the face plate by
two long bolts passing through the cylinder to a rectangular bar of steel pressed
against the other plane end ; the outer surface was then turned. The inner and

480 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

outer surfaces were thus practically concentric, and the ends at right angles to the
axis. The turning was necessarily slow, more than five weeks being occupied on each
of the large cylinders ; the winding of the coils was, however, completed in a few
hours. Alternate cuttings were made of the spirals, of which the grooves were
V-shaped, with an angle of 85°, and of -j-8- inch pitch. It was very important that each
groove should be midway between its neighbours, and the lathe was operated to effect
this ; subsequent microscopic examination proved the equality of distance. While in
the lathe, the diametral uniformity of the grooves was tested by winding in different
parts of the cylinder a couple of turns of No. 24 copper wire, and estimating the
difference in the diameters of the various turns by the touch of callipers. The
cutting tool used was hardened in mercury and was not tempered.

On each large cylinder there are two pairs of coils, the central portion being left
unwound for an axial distance of -fa of an inch. The leads of all the coils must lie in a
plane containing the axis of the cylinder, or otherwise the current through them will
exert a force on the current in the suspended coils of the balance. To ensure the
absence of such a force, the following scheme was adopted for the winding of each coil
(see fig. 8, p. 473). From the outside of the cylinder and near one end, two radial cylin-
drical holes, a and b, were drilled ; these are -£± inch in diameter, -3-2- inch from centre
to centre, and 1^ inches in depth ; they lie in a plane containing the axis of the coil.
From the inside surface two other -6-4-inch holes, c and d, were drilled to a depth of
l£ inches in the same axial plane as the others ; one of these, d, is near the centre of the
cylinder, and the other, c, is -3A2- inch from b, the innermost of the previous ones. The
holes a and b have slotted brass nipples, shown in section, screwed into them, and c
and d are bushed on the inside of the cylinder with ivory pieces. After these bush
pieces were screwed into position, the fine radial holes passing through the nipples, the
ivory, and the marble, were drilled ; the diameter of these holes is 0'024 inch, and they
admit of the free motion of a straight piece of No. 24 wire. The radial holes were
drilled in the following manner : — A bar of steel, 2 inches square and 30 inches long,
with two opposite planed surfaces, was bolted to the slide rest so that it projected
towards and was perpendicular to the axis of the lathe ; a ^-inch hole was then
drilled through the far end of it. Into this hole a spindle was fitted, and on the
spindle a small pulley was fixed, so that the whole could be driven by a motor. The
bar was then turned into an axial position, the spindle set parallel to the face plate,
and the radial holes drilled by a fine drill fitted in the spindle head ; the feed was
governed by the pressure of the hand. A check on the accuracy with which the
holes were drilled was obtained in the following manner : — Adjacent helices are
supposed to start in the same diametral plane and at an angular distance of 180°
apart ; the prolongations of the holes a, 6, c, d should therefore be in line with those
drilled for the leads of the adjacent coil. To test this, a straight piece of No. 24
wire was passed through corresponding holes, and pulled taut ; there was no undue
friction, and a centre finder indicated that the wire cut the axis of the cylinder. As

A NEW CURRENT WEIGHER, ETC. 481

each coil consists of a whole number of turns, there was no necessity to rotate the
cylinder from the time the drilling of the first radial hole was commenced to the
completion of the last.

An estimate was made of the accuracy with which the number of turns is known.
On the fixed cyclinders there are 90 turns to each coil and the diameter is about
33'0 centims. From observations on the radial holes, the number of turns is con-
sidered to be correct within 2 parts in 1,0.00,000.

Between the inner orifices of the passages b and c, fig. 8, a short V-groove e, % inch
deep, was cut, and between the corresponding apertures of a and d a groove f f,
-Jfi inch deep, was made ; in these grooves portions of the leads of the coils were laid.

The copper wire with which the four coils were wound was supplied by the
London Electric Wire Company, Limited, on bobbins of the same diameter as the
cylinders. It is hard-drawn bare No. 24 S.W.G., and has a conductivity such that
1 metre weighing 1 gramme has a resistance of 0'149 ohm at 150-5 C. The mean
diameter of the wire is 0'559 millim. ; this is the average of several hundreds of
measurements, the maximum variation being 1 per cent.

As a guide in winding, an arm was fixed to the saddle and tool carriage of the
lathe which supported the bobbin and a small grooved brass pulley over which the
wire passed on its way to the cylinder. At the commencement the pulley was set in
position for a straight feed and the tool carriage was placed in gear with the leading
screw. On the axle carrying the bobbin a grooved pulley was fixed, and around this
a rope passed ; one end of the rope was attached to a spring balance fixed to the
lathe saddle, and the other end was tied to a heavy weight which just swung clear of
the floor. The effective load on the wire during the winding of the coils was 10 Ibs.,
which resulted in an extension of 0'16 per cent., the limit of elasticity not being
exceeded. YOUNG'S modulus for the material of*the wire was experimentally deter-
mined as ri6x 10l> (C.G.S. units). The coefficient of linear expansion of copper is
l'7xlO~4; hence for an increase in temperature of 80° C. the expansion is 0'14 per
cent. When the coils were immersed in paraffin wax the temperature of the copper
was very nearly 100° C., but the wire appeared to be quite taut on the cylinders.
The reason for this is apparent.

In the case of the fixed cylinders the winding was commenced by threading a free end
of the wire from the outside through the hole c and back through b, fig. 8. When a
few centimetres had been pulled through the nipple, it was passed through the slit
therein and pressed back towards the surface of the marble; it was then given a
couple of turns about the nipple and soldered to it. The wire was afterwards pulled
taut and the necessary bends made to commence the winding. The position of these
bends was estimated beforehand and the wire in the vicinity softened over the flame
of a spirit lamp. During the winding the cylinder was rotated very slowly and
stopped after each revolution for a couple of measurements to be made of the diameter
of the wire. On its way from the bobbin to the marble the wire passed between two

VOL. ccvii. — A. 3 Q

482

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

pads of silk moistened with alcohol and afterwards between two pads of dry silk.
From time to time the strands were examined with a lens, but nothing unsatisfactory
was observed. When the winding was complete except for about 10 centims. of the
last turn, the lathe was stopped, and a long U-shaped clamp slipped over that end of
the cylinder farthest from the face plate ; a grooved piece of ebonite was placed over
the last strand and the latter clamped between the ebonite and the marble ; this

Fig. 9. Method of supporting the suspended cylinders during turning.

enabled the tension on the wire of the bobbin to be relaxed. A length of wire
sufficient to complete the connections was then measured off, the free end passed
through the radial hole d, fig. 8, along the V-groove/y, through a, and finally secured
to the nipple by soldering. Throughout these operations the wire was kept as taut
as possible. On the completion of the helix the clamp was
!®{ removed, the V-grooves filled in with paraffin wax, and the

cylinder wrapped round with silk. It was then removed for
diametral and axial measurements.

The turning «f two small cylinders for the suspended coils was
completed in a manner very similar to that described for the
large cylinders, but as the ellipticity of these was comparatively
great, they were rejected. Two other cylinders were chosen and
their inner and end surfaces turned as in the previous cases, but
their outer surfaces were turned between dead centres. The
arrangement is shown in fig. 9. A, B, and C are tight-fitting
collars on the mandril M ; the outer collars have shoulders, and
these and the collar B are turned so as to be a good fit in the
cylinders, which are clamped between A and C by means of three
bolts. The outer surfaces of the cylinders were finished in this
way, and afterwards the double spiral grooves were cut; the
result is very satisfactory.

The connections to the suspended coils (fig. 10) are much

Fig. 10. Part section of
suspended cylinder
showing leads to coil.

simpler than those for the fixed coils. The terminating nipples are placed inside
the cylinder and one-half of each is cut away where it projects from the marble ; the

A NEW CUKEENT WEIGHER, ETC.

483

portions which project are thus half-cylinders : they have axial grooves into which the
leads are soldered.

After winding the coils the traces of fourteen axial planes at equal angular distances
apart were marked on the end faces of each fixed and of each suspended cylinder and on
the ungrooved portions of the outer cylindrical surfaces. A number of holes for the
fixing of spirit levels and sighting pieces were also drilled.

TABLE I. — Observations for Axial Length of Coils on Suspended Cylinder No. 1.

Temperature = 15° '50.

Number of turns.

Axiiil length of

Mean of values in

Calculated axial

N.

N turns.

Column 2.

length of 184 turns.

oentimi.

centinu.

oentimi.

184

12-9838

12-9833,

12-9833,

29

27

36

41

29

39

27

163

11-5013

11-5008,

12-9825,

10

02

08

149

10-5137

10-51340

12-9829,

29

39

31

135

9-6260

9-62545

12-9828,

68

•.

49

51

121

8-5374

8 -63760

12-98280

80

66

84

Mean

12-9829«

The turning of the marble cylinders was very ably done by Mr. TAYLEBSON, of the
Engineering Department of the National Physical Laboratory.

Axial Length of the Coils. — The axial length of each helix was computed from a
large number of measurements ; some of these were made on the complete helix and
others on portions of it. In addition, the mean value was checked by observations on
a steel cylinder, on which a fine spiral groove was cut of the same pitch as the coil.

3 Q 2

484 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. F, SMITH :

The steel and marble cylinders were turned at or about the same temperature, and
the same portion of the leading screw of the lathe was used. For the axial measure-
ments of the coils a cathetometer was employed, but the observations are subject to a
greater probable error than the generality of high-precision cathetometer measure-
ments owing to the boundaries of the wires being somewhat ill-defined. The axial
length of the helices traced on the steel cylinder was determined by a simple
comparator, and the value thus obtained is associated with a very small probable
error. Table I. contains the results of- the measurements on the coils of suspended
cylinder No. 1. The first set of measurements was made on the whole number of
turns, viz., 184; the next observations on 163 turns chosen in various parts of the
coil, and the third, fourth, and fifth measurements were on 149, 135, and 121 turns
similarly chosen. Each of the values recorded in column 2 is the mean of at least
four readings; in all, about 100 observations were made.

If equal weights are given to each set of observations, the mean of the values
recorded in the last column is 12 '9829 centims. The observations on the outer end
wires are not quite so reliable as those on intermediate ones, for a little irregularity
is always possible when starting and finishing a winding ; eight observations (each
being the mean of four) are therefore included in the first set. The mean of eight
measurements on the steel cylinder is 12'9830 centims., a much closer agreement
than was anticipated. Taking the value 12 '9 829 centims. and the values recorded in
Column 4 of Table I., the differences (observed — mean) are +4, — 4, +1, —1, and — I//.,*
from which a probable error of O'OOl per cent, is deduced if we exclude the error of
the gauges employed. The observations on the helices of the other three cylinders
are equally satisfactory, and the means of the readings obtained with them are given
in Table II.

As the tool carriage travelled over different portions of the leading screw of the
lathe when cutting the spiral grooves in the suspended cylinders 1 and 2, the
uniformity of the screw was tested and an estimate of 2/A was made as the probable
difference in length of the coils on the two cylinders, that of the coils on No. 1 being
the greater. The recorded measurements show that No. 2 is probably the longer by
this amount, the values being 12'9829 centims. for No. 1 and 12'9831 centims.
for No. 2. On the whole, the observation error of the axial lengths may be taken as
of the order l£ parts in 100,000.

For the diametral measurements a machine, shown in fig. 11, was obtained from
Messrs. STANLEY. This consists of a double-webbed rectangular steel girder, two
micrometer heads, and various supports for gauges, &c. To each of the micrometer
heads an optical lever of the form shown in fig. 12 was attached. A well-fitting
hardened steel piston P is tapered and ground at one end so as to form a plane edge
^ inch wide and -^ inch deep ; the other end tapers more slowly and terminates in a
rounded end -^ inch in diameter. This end of the piston fits into a rectangular

* p. = 1 micron, or pg^ of a millimetre.

A NEW CITRltKNT U Kit ;ll H;. ETC.

485

TABLK II. — Observations for Axial Length of Coils. Temj)erature 15C>50.

Cylinder under
observation.

N — Number
of strands
observed.

Axial length
of N
strands.

Total

number
of oboer-
vations.

Total

till MS.

Calculated
axial length of
coil.

Difference
from
mean.

Suspended cylinder
No. 2

184
163
149
135
121

cfntin -
12-98340
11-5014,
10-51338
9-52530
8-53800

32
12
12
12
12

184

'•rlltilllv

12-9KU,,
320
297
26,
34,

+ 3/i
+ 1
-2
-5
+ 3

Mean- 12-9831,

Steel cylinder . . .

—

8

—

12- 9*28

Fixed cylinder No. 1,1
upper portion

163
149
135
180

11 -5018s

10-51360
9'52584
12-70070

If,
16
16
8

180

12-7014,
099
111

070*

+ 3M

0
_ ^

M.-iu, 12-7011,

Steel cylinder . . .

—

—

8

—

12-7008

Fixed cylinder No. 1,1
lower portion

163
149
135

. 180

11-5022,,
10-51380
9-5260,,
12-70070

16

16
16

8

ISO

12-7018.
12,
H,
070*

+ 5/1
-2
0

-7

Mean =12 -7013,

Steel cylinder . . .

—

—

8

—

12-7008

Fixed cylinder No. 2, 1
upper portion

163

149
135
180

11 •:."]. ;.
10-5135.
9-5255,
12-70140

16
16
16

8

180

ll>-701 1;,

096
077
14*

* I/'
-1
-3

+ 4

Mean = 12 -7010,

Steel cylinder . . .

—

—

8

—

12-7008

Fixed cylinder No. 2, 1
lower portion

163
149
135
180

11-5016&
10-5140i
9-5258o
12-7014«

16
16
16

8

180

12-7012,
154
10r
14*

"/'
+ 3
-2
+ 1

Mean = 12-7012..,

Steel cylinder . . .

—

—

8

—

12-7008

* These values were determined by measuring the complete axial length of the upper and lower coils
plus the central gap.

Observations on 1G3, 149 and 135 strands indicate practical equality of the axial lengths of the upper
and lower helices and also enable the length of the central gap to be calculated. This latter length was
subtracted from the total and the result divided by two. In taking a mean of the values given in
Column 6, only half the weight has been attached to the " * " observations.

48«

PROFESSOK W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

groove cut in a brass bar carrying a plane mirror M ; the bar is free to rotate about a
steel axle, its movement being in a vertical plane. A comparatively strong brass
spring on each side of the bar causes it to continually press against the piston if the
rounded end of the latter projects beyond the stop S. The vertical distance between

1

Fig. 11. Machine for measuring diameters of cylinders.

the axis of the mirror and the contact point of piston and brass bar is about 1*5 millims.,
and the height of the micrometer head above the upper surface of the girder is
30 centims. When the barrel B of the micrometer advances, the piston, mirror,
bearings, &c., advance with it until the plane end of the piston comes in contact with
a rigid body ; an advance of the micrometer barrel then results in a tilt of the mirror

Fig. 12. Optical lever for measuring diameters of cylinders.

nd the consequent deflection of a spot of light. All the parts are well made and the
bearings lubricated with clock oil. A Nernst filament was used as a source of light,
and a lens produced a sharp linear image on a white surface 150 centims. distant from
the mirror. A forward motion of the micrometer barrel of I/A resulted in a deflection

A NEW CURRENT WEIGHER, ETC. 487

of 2 millims. on the scale ; a difference of I/* was thus read with ease. As there was
no silk on the wire to interfere with precise measurement, this high sensitiveness was
well worth attaining. The contact planes of the pistons were tested for parallelism
with satisfactory results ; tests were also made which indicated that these planes were
normal to the axes of the pistons.

To facilitate the diametral measurements, the marble cylinder under observation
was supported on a turn-table provided with ball bearings and levelling screws
(fig. 11). The original intention was to support this table on a separate platform and
so avoid the bending of the girder which results when it supports the load ; this,
however, proved to be unsatisfactory, and ultimately a small wooden platform was
bolted to the girder, and on this the turn-table and cylinder rested. The traces of
the axial planes on the ends and ungrooved portions of the cylinder, and the knowledge
that the ends were at right angles to the axis, enabled the coils to be rapidly set in
position so that their axes were verti«al ; at the same time the adjustment ensured
that the plane edges of the touch-pieces would come into contact with the copper
wires at opposite ends of a diameter. Two spirit levels at right angles were used for
the levelling of a cylinder, and it was usually found necessary to make a slight
adjustment for every measurement made in a different axial plane. In general,
observations were made in eight approximately equidistant diametral planes, and in
each of these, 14 measurements were taken in equidistant axial planes; at the
conclusion of the 14 observations the first was repeated as a check on the constancy
of the apparatus. The method used was not a " null" one ; the zero reading, i.e., that
when a mirror was against a stop, was observed from time to time, and a constant
deflection of 10 millims. from this was adhered to throughout the measurements. The
apparatus worked very smoothly, the readings being easily reproduced to I/*, and
only in a few cases of uncertainty was more than one observation made of any one
diameter. The temperature of the room was very nearly constant and equal to
150<5 C. ; a Richard's thermograph recorded the variations.

At the commencement of a series of measurements, tlie Whitworth steel gauge
(square section, flat ends) was placed in position, and the uprights carrying the
micrometer heads were bolted to the girder. A mass equal to that of the turn-table
and cylinder was next placed on the small platform between the micrometer heads,
and the observations on the gauge were then made, the latter being displaced and
reset between every two measurements. The cylinder was then placed in position
and measured, and afterwards the gauge was again set up. With respect to the
latter measurements, the difference in the readings of the micrometer heads never
varied by more than 2/i from the commencement to the completion of a series of
observations. When measuring a diameter, the touch-pieces made contact with one
wire of each helix and the mean of the observations gave, therefore, the mean outside
diameter of the two coils. To determine the difference of the mean diameters of the
coils, one of the micrometer heads was raised -fa inch and a few observations of

488

PROFESSOR W. E AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

difference made ; the value of this difference varies from 2/t to 3/x for the different
pairs of coils. A confirmation of this difference appears on p. 516.

In Tables III. to VI. there are given the diametral measurements of the coils in
various planes, the mean diameter of the wire with which the coils are wound, and
the mean diameter of the coils to the central filament of the wire. In the

TABLE III. — Results of Measurements of the Diameters of the Coils, to Centres of
Wires, on Suspended Cylinder No. 1. Temperature, 15° '5 C.

Containing strand number —

Diametral plane
number.

4

38

68

80

92

104 116

146

180

= 20 • 3500 centims. +

1

79/x

79/*

80/x

80/i

81/i

83/i

84/x

91/i

94/t

2

77

78

84

82

81

83

84

94

94

3

75

77

80

83

83

83

86

92

96

4

76

74

80

79

82

84

85

90

93

5

75

77

79

79

78

81

85

91

96

6

74

77

79

82

82

83

86

89

93

7

76

78

79

79

81

83

86

86

91

8

77

78

81

80

82

82

87

89

90

9

76

80

82

82

82

84

86

87

89

10

80

80

82

82

81

86

88

84

92

11

80

77

82

82

83

86

84

86

90

12

78

82

83

84

84

85

86

88

92

13

80

82

83

82

84

87

86

90

90

14

78

80

83

80

83

83

86

90

91

Mean . .

772

785

81|

81i

Bit

838

856

89]

922

Number of observations made to determine the mean diameter of the wire, 46.

Greatest difference between any two observations, 0 • 8 per cent.

Mean diameter of wire, 0 • 559 millim.

Approximate mean diameter of coils = mean of the values in table = 20 • 35834 centims.

Mean diameter of coils, computed from calibration curve = 20'35835 centims.

Difference of diameters of neighbouring convolutions = 0 • 0003 centim. (approx.).

conversion of inches to centimetres the ratio 2'539998 has been taken. For the data
relating to the steel gauges employed we are indebted to Mr. ATTWELL, of the
Metrological Department. A good conception of the ellipticity and conicality of
the coils is afforded by the calibration curves which follow (figs. 13, 14, 15, 16). The
suspended coils are very slightly elliptical, and the conicality is also very small and
uniform. The difference in the extreme mean diameters of suspended cylinder No. 1
is 17/1, equivalent to an average slope of 1 in 8000; the corresponding value for

A NEW CURRENT WKHWKR, KTC.

489

suspended cylinder No. 2 is about 1 in 10,000 ; the larger ends of these cylinders are
the ends which were in contact during the turning. The ellipticity of the coils on
the fixed cylinders is greater than that of" the suspended coils, but it is much too
small to influence the calculation of the mutual induction, for the variation of mutual

PLANE Of MEASUREMENT
S 6 7 S 9 IO II

IZ IS

38

80

•

"e

85
•o

ISO

*

20^
*

ELLIPTIC! T Y

Fig. 13. Suspended cylinder No. 1.

induction with small changes in radius is approximately linear. For the same reason
the conicality of the coils may be neglected. The larger end of each fixed cylinder is
that which was secured to the face plate of the lathe during the final turning and
screw cutting.

VOL. OCVII. — A. 3 R

490

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

TABLK IV. — Results of Measurements of the Diameters of the Coils, to Centres of
Wires, on Suspended Cylinder No. 2. Temperature 15°'5 C.

Containing strand number —

Diametral
plane number.

4

38

68

80

92

104

116

146

172

180

= 20 • 3500 centims. +

1

M*

86/t

87/i 88/1

88/1

85/x

87/i

87/i

96/i

97/i

2

86

88

90 87

87

85

88 91

94

95

3

85

80

88

88

87

87

92

92

95

97

4

84

86

88

86

88

87

92

91

96

95

5

84

85

86

85

87

88

88

86

96

97

6

85

87

88

88

88

90

91

92

94

95

7

85

83

88

89

88

90

91

94

96

100

8

85

86

85

87

90

90

93

93

97

98

9

84

88

86

84

90

88

90

94

97

98

10

84

86

84

85

88

88

90

91

97

96

11

84

87

86

85

88

90

88

91

97

96

12

83

87

87

88

84

90

89

94

96

97

13

83

89

90

89

87

90

92

92

96

97

u

85

87

89

89

87

86

88

89

95

95

Mean . . .

84,

86,

878

870

87«

881

89»

91a

959

'.Hi,,

Number of observations made to determine the mean diameter of the wire,

46.

Greatest difference between any two observations, 0 • 8 per cent.

Mean diameter of wire, 0 • 559 millims.

Approximate mean diameter of coils = 20 • 3589,-, centims.

Mean diameter of coils, computed from calibration curve = 20 • 35890 centims.

Difference of diameters of neighbouring convolutions = 0 • 0002 centim. (approx.).

The following convention is adopted in numbering the 14 axial planes. The upper
plane end of a cylinder is viewed and one of the two marked diameters nearest in line
with the connectors of the coils is called No. 1. The direction of ascending numbers is
clockwise, diameters 1 and 14 being on opposite sides of the plane containing the leads.

An idea of the probable error of the mean diameter of any one coil may readily be
obtained. The values of the standards of lengths employed are known in terms of
the National Physical Laboratory 12-inch end gauge of similar type standardised by
the Board of Trade. The absolute values are not of importance, however, for if the
dimensions of the fixed and suspended systems change in the same proportion and in
the same direction, the force due to the current is unchanged. For the smaller coils,
an 8-inch steel gauge was used ; for the larger ones, this was combined with a 5-iuch
gauge. The ratio of the lengths of these gauges was known with an error certainly
less than 5 in 1,000,000. The probable error due to the setting of the gauges in the

A NEW CURRENT WKIQHKK, KTC.

491

measuring machine bed was much greater than this, but an analysis of the readings
leaves little doubt that the prolwble error is not more than 1/x. On each jwir of coils
not less than 112 observations were made, and the curves show that the error of a
le olwervation must l>e small ; hence the error of the mean diameter deduced from

PLAMf. OF MCASUR£M£NT

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CLLIPT/C/TY.

Fig. 14. Suspended cylinder N<>. •_'.

the 112 observations is not appreciably greater than the error of the gauge. We
conclude, therefore, that the relative diametral dimensions of the coils are correct to
5 in 1,000,000. The probable error of the axial lengths given in Tables I. and II. is
of the order 1 5 in 1 ,000,000, and the calculated value of the mutual induction should

3 R 2

492

PROFESSOR W. K. AYRTON, MR. T. MATHER AND MR. F. K. SMITH:

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Fig. 15. Fixed cylinder No. I.

4U4

PROFESSOR W. E. ATRTON, MR. T. MATHER AND MR. F. E. SMITH:

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Number of observations made to determine the mean diameter of the wire : 44 (upper), 44 (lower).
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Mean diameter of wire = 0-559 millim. (upper), 0-559 millim. (lower).
Approximate mean diameter of coils = 33'00:>7S centims. (upper), = 33'0039i centims. (lower).
Mean diameter of coils, computed from calibration curve = 33 -0028s centims. (upper), 33-00396 centims. (lower).
Difference of diameters of neighbouring convolutions = 0-0002 centim. (approx.).

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496 PROFESSOR W. E. AYKTON, MR. T. MATHER AND Mil. V. E. SMITH:

be correct within about 5 in 1,000,000. This conclusion assumes absolute constancy
of the dimensions of the coils in the interval between measurement and erection, or
the same relative change in dimensions. The values of the mutual induction of the
coils on fixed cylinder No. 1 and on suspended cylinder No. 1, and of the coils on the
No. 2 cylinders were independently calculated by two. of us (T. M. and F. E. S.) in
July, 1905 (see Section 10), the difference in mutual induction of the two systems, as
calculated, being 0'0062 per cent., that of the first system being the greater. When
the ampere balance was completed and the equipment and settings made satisfactory
(September, 1905), the difference in mutual induction as found experimentally was,
and still is (April, 1907), 0'0054 ± 0'0004 per cent., that of the first system being
the greater. Particulars of this experimental determination will be found on p. 515.
An experimental estimate of the difference in mean diameters of two coils on a
suspended cylinder is 3'5/u, + l/i (p. 516), a value in satisfactory agreement with the
difference found by direct measurement.

Insulation of Coils. — The insulation of the helices was next proceeded with. For
this purpose an X -shaped framework of wood was mounted on a metal axle and fitted
inside the marble cylinder under observation ; the axle was supported on bearings, so
that the whole could rotate freely. The marble is semi-transparent, and when an
electric lamp is placed inside a cylinder the air gaps between the strands are easily
inspected. Under these conditions the appearance of the coils was very beautiful,
and close inspection with a lens failed to reveal any defects in the winding. The first
measurement of the insulation resistance between adjacent helices indicated it to be
of the order of 50 ohms, and the filament of an electric lamp glowed brightly when
placed in a circuit containing the two helices and the gaps of what appeared to be
marble and air. That the marble was not at fault was shown by tests on the
unwound portion of the cylinder, and examination of the gaps with a powerful lens
failed to reveal any metallic bridging pieces. In their shortest parts the gaps are
0*15 millim. long, and on the fixed cylinders there are four gaps, each nearly
93 metres wide ; several days were spent in their examination, and on one occasion a
silk thread was passed between neighbouring strands ; the insulation resistance still
remained less than 100 ohms. It is unnecessary to describe in detail all the subsequent
attempts to locate the leaks. The cylinder and coils were washed with a thin shellac
varnish, made by dissolving shellac in ether, but there was no improvement ; after-
wards they were washed in ether and then absolute alcohol, but without noticeable
effect. The cause of the low insulation resistance was apparent, however, for at the
bottom of a porcelain dish containing the used alcohol a fine sediment settled which
consisted of minute particles of copper. Apparently the copper strands had a very
loose, scaly skin, and thousands of these tiny particles of copper were bridging the air
gaps and so diminishing the insulation resistance. The washing with alcohol was con-
tinued and the strands lightly brushed with a camel-hair brush, a 32-c.p. lamp being
lit through the circuit containing the gaps. Eventually two 32-c.p. lamps were placed

A NEW CURRENT WEIGHER, ETC. 497

in parallel, so that the current was about 2 amperes. After the washing with alcohol
had l>een continued for 20 minutes or half-an-hour there was a crackling noise, and
hundreds of tiny sparks appeared over the surface of the cylinder ; simultaneously the
lamps ceased to glow. A measurement of the insulation resistance between the coils
showed it to be of the order of 300 megohms ; the shorting pieces had been burnt out
with a most satisfactory result. To prevent the recurrence of the low insulation resist-
ance the washing was continued ; occasionally the lamps glowed, but with continued
washing the shorts were burnt out as before. When the insulation resistance was of
the order of 1000 megohms, with an applied pressure of 20 volts, the cylinder was
lifted from its bearings and placed in others secured to a framework resting on the
top of a bath of melted paraffin wax. About one-third of the circumference of the
cylinder dipped into the hot liquid. The cylinder was rotated until the marble was
sufficiently warm to keep the wax on its surface in a liquid condition ; it was then
removed for the wax to solidify, and afterwards dipped once more, in order to obtain
a thicker coat. The insulation resistance was measured while the cylinder was hot,
and also when the wax had solidified ; the latter value was always the greater.
After the lapse of a week or ten days, the ends and interior of the cylinder were
cleaned and preparations made for further measurements of diameters. The wax was
carefully removed from several parts of the cylinder and the strands cleaned by
rubbing with a small pad of silk ; the measurement of six or eight diameters was
then carefully made, the steel gauges being set up as before. A summary of these
measurements follows (see Table VII.), from which it is inferred that there was no
appreciable change in the diametral dimensions.

In. one of the large cylinders the insulation resistance between the two upper and
the two lower helices was at first comparatively low, viz., 2000 megohms. The cause
of this .was found to lie in the internal ivory plugs through which the copper leads
passed. As it was impossible to remove these without stripping the cylinder, they
were slotted in such a way as to reduce theTsection of the conducting material ; the
insulation resistance was thus increased to 10,000 megohms. Insulation tests on
fresh ivory pieces were invariably satisfactory, but two such pieces inserted in the
ampere balance appeared to deteriorate with time, and eventually had to be replaced
by ebonite.

To prevent damage to the surface of the wax with which the coils were coated, it
was thought desirable to cover it with a harder insulating material. Shellac varnish
was tried and used for the larger coils, but the suspended ones were untouched owing
to the results of experiments on equal surfaces of paraffin wax and shellac varnish.
The latter was found to be much more hygroscopic than the -former. From the
measurements made it is estimated that each suspended cylinder coated with paraffin
wax would change in mass by 6*8 milligrammes if removed from a dry atmosphere
to one saturated with moisture ; had the outer coating been shellac varnish the
corresponding change would be 146 milligrammes,

VOL. covil. — A. 3 8

498

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

TABLE VII. — In which the Diametral Measurements on all the Coils, before and
after Insulating with Paraffin Wax, are Compared.

Coil.

Diametral plane
number.

Strand
number.

Diameter before
insulating.

Diameter after
insulating.

Difference.

centims.

eentims.

1

38

20-3579

20-3576

-3,x

1

9

92
38

81
80

80
80

-1
0

9

92

82

80

-2

Mean difference =

-Is

c

1

4

20-3584

20-3586

+ 2/i

1

80

88

86

-2

Suspended No. 2 . . . •

1

9

104
4

85
84

84

84

_ i

0

9

80

84

83

-1

.

9

104

88

89

+ 1

Mean difference =

-03

t

1

175

33-0010

33-0006

-4,1

3

175

16

15

-1

9

175

17

18

+ 1

Fixed No. 1, upper part -

12

1

175
59

10
32-9995

07
32-9994

-3
_ j

8

59

33-0006

33-0009

+ 3

12

59

32-9999

01

+ 2

.

10

59

33-0002

00

-2

Mean difference =

-0,

1

175

33-0008

33-0008

0/x

3

175

17

17

0

10

175

13

10

-3

Fixed No. 1, lower part •

12
1

175
59

09
08

07
07

-2
• -1

8

59

21

22

+ 1

11

^9

' 14

18

+ 4

.

12

59

12

09

-3

Mean difference =

-06

1

170

33-0029

33-0028

-1/i

5

170

27

26

-1

Fixed No. 2, upper part •

8
2

170
125

37

25

34
25

-3
0

5

125

34

33

-1

10

125

37

37

0

Mean difference =

-lo

• r

4

170

33-0025

33-0024

-1/t

6

170

38

41

+ 3

Fixed No. 2, lower part -

14
1

170
125

34
36

33
37

-1

+ 1

6

125

45

45

0

10

125

49

50

+ 1

Mean difference =

+ 0,

A NEW CURRENT WEIGHER, ETC. 499

SECTION 7. — ERECTING AND ADJUSTING THE INSTRUMENT.

To facilitate the setting of the fixed cylinders on the balance table, two spirit
levels and four sets of cross-wires are mounted on the upper plane end of each. The
sensitiveness of the levels is such that a tilt of 20 seconds of arc displaces the air
bubbles 1 millim. from their central position. A Whitworth surface plate was
levelled and on this the spirit levels were set; afterwards a marble cylinder was
rested on the plate, which was then relevelled, and two other levels placed at right
angles on the upper end of the cylinder ; the displacement of a bubble from its mid-
position was practically unreadable, the parallelism of the plane ends of the cylinder
being thus confirmed. The levels were screwed to the cylinder and re-set to read
correctly.

At opposite ends of two diameters at right angles, four slides with upright pieces
carrying cross-wires are screwed to the upper plane end of each cylinder (see fig. 2,
Plate 7). These are adjustable in azimuth and the final setting is such that the line
joining the points of intersection of opposite cross- wires lies in a plane containing the
axis of the coils. The setting was made by suspending a weighted thread inside the
cylinder so as to coincide with the axis, the indicator of thjs adjustment being a
centre finder. A cathetometer telescope was next focussed on the thread and on one
of the cross-wires, and was altered in position until the plane of the vertical wire of
the telescope lay in the same plane as the thread and the vertical cross-wire. The
cross- wires opposite to this latter were then adjusted in azimuth until they also lay in
this plane.

Each suspended cylinder carries a brass T-piece supporting two sjjirit-levels at
right angles ; in addition a tripod is supported which in turn carries a pointed rod
to be seen projecting above the fixed cylinder in fig. 2 (Plate 7). The ends of the
tripod legs enter into the cylinder and are turned to be an exact fit. The rod is
for the adjustment to coincidence of the axes of the fixed and suspended coils ;
it is adjustable in vertical height and its pointed extremity lies in the axis of the
coils ; it is set so that when its extremity is in the plane of the cross-wire inter-
sections the suspended and fixed coils are symmetrical as regards vertical height.
The coils are concentric when the lines joining opposite cross-wires intersect in the
axis of the rod.

Concentric cable is used for the leads to and from the various coils. The junctions
of the cable with the fixed coils are shown in fig. 8, and those with the suspended
coils in fig. 10. In the case of the fixed coils the ends of the wire leading to any one
of the coils were first soldered to small brass blocks supported by a strip of ebonite
which in turn was screwed to the cylinder ; the ends of the leads of the concentric
cable were similarly soldered to two small brass pieces which were screwed in contact
with those leading to a coil. The cable could thus be easily removed without in any
way damaging the connecting pieces. In the case of the suspended coils, the wires

3 8 2

500 PEOFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

leading from them terminate at the brass connectors inside the cylinder. These
connectors are grooved, and into the grooves stout pieces of copper wire are soldered
and lead directly to the concentric cable. The junctions are shown in fig. 10, p. 482.
To take part of the weight of the cable attached to the small cylinders and thus
prevent the connections from being strained, two small curved arms project from
each suspended cylinder, and to these the cables are clamped. They may be seen just
above the fixed cylinders in fig. 2, Plate 7. Each small cylinder is suspended by three
phosphor-bronze wiresw, w, w, fig. 6, Plate 8, attached to a three-limbed spiderS ; to these
wires the cylinders are hooked by brass strips screwed to the interior of the cylinder
and bent at right angles at their lower ends ; the feet thus formed fit into recesses
cut in the marble. The effective length of the phosphor-bronze wires is adjustable,
and by such adjustment, together with an alteration in position of the heavy nuts on
the limbs of the spider, the cylinders are levelled. On the completion of the sus-
pended coils and their fittings, the mass of one suspended system was different to that
of the other by 2 grammes in a total of 5500 grammes ; equality was obtained by
loading one of the T-pieces.

Above each suspended cylinder a commutator C (fig. 17) is supported by one arm of
the three-limbed spider. The concentric cables from the coils pass to this commutator,
and from the latter two bare copper wires, shown black, are taken to an ebonite block B.
A second ebonite piece B' is screwed to the main pillar of the balance, and between
B and B' 160 silver wires are suspended ; the diameter of a single wire is 1 mil (25/u.).
A long length of concentric cable completes the circuit to a multiple commutator and
plug board. By appropriately setting the commutator C, the current can be made to
circulate in the same or in opposite directions in the two helices, fig. 3, and by
suitable connections to the multiple commutator and to the commutator C the
insulation resistance between the helices can be measured. Wherever possible, the
non-concentric leads to and from the coils are kept very short and placed radially or
parallel to the axes of the cylinders ; also, the feed and return leads are placed as
close together as practicable ; the design thus ensures the minimum of force between
the current in the fixed coils and that in the commutator and leads to the suspended
.coils. The commutator C is a simple one of four copper quadrants with a turning
head of ebonite, carrying two contact pieces ; these latter are insulated from each
other and are attached to the ebonite head by hard springy copper ; they are split
midway to ensure uniform pressure on all the quadrants when the turning head is
correctly set. The commutator can reverse the current in one of the helices only,
but the concentric leads from both coils pass to the commutator block ; this is for
convenience in making the connections, and to obtain symmetry of distribution of the
current leads. By making one of the contact pieces (say Q, fig. 17, a) slightly longer
than the other, the commutator may be set in position suitable for making the
insulation test between the two helices.

The 160 silver wires are divided into two portions, which are insulated from each

A NEW CURRENT WEIGHER, ETC.

501

other ; in any one division the 80 strands lie in two parallel layers, the system being
formed by winding a continuous length of silver wire over two brass rods cut with a
screw thread of ^g-inch pitch ; no two strands are in contact. After the completion
of the winding the silver wires were soldered to the rods by running a very soft

Fig. 17.

Leads and commutator for suspended coil, a shows diagrammatic plan of commutator

(looking upwards).

fusible metal into the V-grooves. The length of a strand is 10'5 centims., but the
distance between the rods is a little less than this. Fig. 17 illustrates the manner in
which a set of silver wires is placed in position. Before their insertion the sensitive-
ness of the balance was such that an added load of 1 milligramme produced a
deflection of 0'9 scale division ; afterwards the corresponding deflection was 0'85
scale division, a diminution in sensitiveness of 6 per cent. only. The concentric
cable attached to B' is clamped to the main pillar of the balance, and passes through
a hole drilled in the base of the balance table to the multiple commutator and plug
board described on p. 521.

The first setting of the coils on the tables of the balance involved the following
operations :—

(a) Levelling of one suspended cylinder by adjustment of the lengths of the
suspension wires and of the masses on the limbs of the spider.
(6) Levelling of the concentric fixed cylinder.

(c) Approximate setting of the fixed cylinder, so that the mean diametral plane of
all the coils on it coincided with the mean diametral plane of the suspended coils.

(d) Levelling and vertical adjustment of the other fixed cylinder until its mean
diametral plane coincided with the corresponding plane of the first fixed cylinder.

(e) Levelling and adjustment in vertical height of suspended cylinder No. 2 in
order that conditions (a) and (c) should hold with it.

(/) Setting of each pair of cylinders to be concentric.

502 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

This mechanical method of setting the coils is subject to errors which may be
serious in an instrument intended for observations of high precision. So far as we
know, no attempt has hitherto been made to set two coaxial coils in a position
of maximum force by an electrical method. The ampere balance lends itself to such
a setting, and the accuracy thereby attained is considerable.

ELECTRICAL METHOD OF SETTING THE Cons.

(1) Setting to Coincidence of the Mean Diametral Plane of the Suspended Coils
with the Corresponding Plane of all the Coils on the Concentric Fixed Cylinder. — If
MB is the difference of the mutual induction of the upper fixed coils and the
circular ends of the suspended coils, and M, that of the lower fixed coils and the
same, and if the currents flow in opposite directions in the upper and lower fixed
coils, the force between them and the suspended system is yhy (MB4-M(), where yh is
the current through the fixed coils and y is the current per unit axial length in the
current sheet equivalent to the current in the suspended coils. This is the maximum
force possible for the coaxial system, and variations in the force for small axial
displacements are also small. The rate of change was determined by passing a
current of 1 '02 amperes through all the balance coils, the direction of the current in the
various helices being such that the two suspended systems were subject to the maximum
axial forces, but opposed to each other so that the total turning moment on the beam
was small and almost nil. One set of fixed coils was now displaced through known
axial distances and the change in the resting point of the balance observed ; that
position of these fixed coils when the force due to them is a maximum is the correct
axial position for minimum mutual induction. The results obtained with one of the
systems are plotted in fig. 18 ; for such displacements as those made the force is
approximately given by the expression : — maximum force multiplied by (1 — 1 1 x 10~8cP),
where d is the displacement in mils and is measured from the plane of minimum
mutual induction. The force may also be written : — maximum force multiplied by
(1 — O'OlTx3), where x is the displacement in centimetres. For a displacement of
10 mils (254/n) the change in force is 11 in 1,000,000, which for a current of 1 ampere
is equal to 0*04 dyne approximately.

There is, however, another method of setting the cylinders which is even more
sensitive. If, instead of the currents flowing in opposite directions in the upper and
lower fixed coils, they flow in the same direction, the force between them and the
suspended system is yAy(Mu— M,). When the coils are set in their correct position,
this is nearly the minimum force possible for the arrangement, and the rate of change
of force with axial displacement is large. Observations were made with the current
circulating in this manner in one set of fixed coils, the current in the system on the
opposite side of the balance being so directed that no measurable force was produced
by it. The correct position of the fixed coils in one of the systems is when the

A NEW CURRENT WEIGHER, ETC.

503

force is equal to 0*14 dyne when the current is 1 ampere and circulates in the same
direction through all the coils of the system. The corresponding force for the other
system is 0'34 dyne. The value of the force for a displacement d mils from the correct
axial position when 1*02 amperes is passing is given by the expression 4'8 x lQ~*dg dynes,

MILS

+ 307

2

| +Z07

j

U

o 0
% K+107

r y

w H

5* +7

0 t —43

>• c

5 < -93

h

0

5 2

S; 2-'9s

5 *•

-i
<

* -295
-593,

^^

x^*

s

s*

s\

s^

j

/

/

/

/

f

\

\

~\

\

^

s^

S

^

^.

-^

'

O — lO'O — 2dO — 3OO — 4OO

CHANGC OF FORCE IN OVKJCS (CUR .• I'OS AMP.j

Fig. 18. Change of force due to axial displacement of coils.

where g = 981 centiins./sec*. The force is also given by O'ldxg, where a; is in centi-
metres. Thus, for an axial displacement of 1 mil of the fixed or suspended coils, a
change in balancing mass of 0*48 milligramme results ; the axial position may there-
fore be fixed to less than 1 mil.

(2) Setting of the Fixed and Suspended Coils to be Concentric. — When the coils
are coaxial the mutual induction is a minimum with respect to radial position, and
when the current flows in opposite directions in the upper and lower fixed coils the
force changes with displacement from the coaxial position. The change in force with
radial displacement was measured for both fixed cylinders, the displacements being
made in two directions at right angles ; the results were plotted in four curves, of
which two (those for the- left fixed cylinder) are given in fig. 19. Inspection shows
that the rate of variation of mutual induction with radial displacement increases with

504

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

the value of the latter, and that the position for minimum rate of change of mutual
induction may be deduced with considerable accuracy. The force in every case is
approximately given by the expression : — maximum force multiplied by (1 + 5'8 x lO'V/*),
where d is the radial displacement in mils from the coaxial position ; the corresponding
expression when the displacement is in centimetres is : — maximum force multiplied by
(l + O'OOQx3). Thus a displacement of 10 mils from the coaxial position produces a
change in force of 5'8 parts in 1,000,000. By the aid of the curves the radial setting
can be made within 2 mils, so that the error introduced by faulty radial setting is not
greater than 1 part in 5,000,000.

DIRECTION OF DISPLACEMENT

V)

u

fes

o

z
u

O -*'S

§

u.
u.
0

o

x

0 0-JT

\-f ^ ^

J \J

\

\

[

I

1

\

/

\

I

\

1

\

1

\

/

\

\

/

1

\

/

\

/

^.

— -

s

•*

RADIAL DISPLACEMENT OP LEFT FI*ED CYLINDER (ARBITRARY ZEROS)

Fig. 19. Variation of force caused by radial displacement of coils.

Effect of the Leads. — It was possible that the current in the leads to and from
the fixed coils might exercise an appreciable force on the suspended system, and that
the movable leads connected to the latter might be affected by the current in the
fixed system. This was tested by completing the circuit through the leads only of
the fixed coils and through the suspended system, and noting the effect ; afterwards
the current was passed through the leads only of the suspended coils and through
the fixed coils, and the result again noted. Absolutely no force was detectable, and
on a subsequent repetition of the experiment the same result was obtained.

Insulation Tests. — When making a determination of current, the greatest difference
of potential between any portion of the balance and the earth was about 74 volts, and
the greatest difference of potential between neighbouring strands on the same cylinder
was less than 7 volts. It is desirable, therefore, that the insulation resistance between
the balance circuit and earth should not be less than 100 megohms, and that between
neighbouring strands should not be less than 10 megohms. The insulation resistance

A NEW CURRENT WEIGHER, ETC. 505

of the various parts has been measured on several occasions, and the lowest measured
resistance between any two adjacent coils is 2000 megohms, and between the balance
circuit and earth it is 1000 megohms. The applied pressure in the former measure-
ments was, in general, 40 volts, and for the latter 200 volts. When the coils are
arranged in two groups, so that each group consists of one coil of each pair, the
insulation resistance is 1500 megohms. The first measurement was made in March,
1905, and the last in April, 1907.

SECTION 8. — ADVANTAGES OF DUPLICATING THE COILS.

As previously mentioned, there is a set of coils at each end of the balance. Several
advantages are gained by this arrangement. In the first place, the force to be
measured is doubled by using the two sets of coils, and the accuracy of the measure-
ment is therefore increased. A much greater advantage, however, arises from the
symmetry thus obtained, for mechanical disturbing causes will, on the whole, tend
to be neutralised.

One of the principal disturbances arises from convection currents produced by the
heat generated in the coils, and in the flexible connections to and from the suspended
systems. Another is the change of buoyancy due to change of temperature of the
air in which the suspended coils hang. Both these produce a fairly rapid drift of the
rest-point of the balance when a single set of coils is used, but when both sets are
employed the steadiness of the balance is greatly improved. The extent of this
improvement will be seen on reference to fig. 20, which shows four pairs of curves
taken to test this matter. During all these tests the adjacent helices on each cylinder
were connected up, so that the current (if any) flowed in opposite directions in
adjoining wires, thereby making the windings inoperative, and obviating the necessity
of keeping the current very constant.

Several sets of about 12 readings of the swings of the balance were taken under
each of the following conditions respectively : —

(a) No current through either set of coils.

(6) Normal current through both sets of coils.

(c) „ ,, „ left-hand set of coils.

(d) „ „ „ right-hand set of coils.

The rest-points were calculated from each group of three successive readings
throughout a set, and the values tabulated, thus giving the rest-points for each half
period. From the several sets of observations taken under each of the conditions (a),
(b), (c), (d) respectively, those showing the least and greatest drifts were plotted,
the former being shown in full lines and the latter dotted in tig. 20. The points thus
obtained were joined by straight lines, and no attempt made to smooth out
irregularities. In this figure the middle of the balance scale is denoted by 100 ; one
division of the scale is about -^th of a millimetre (actually 0'095 inilliin.), and as this

VOL. cc-vn. — A. 3 T

506

PKOFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

is represented on the curve by a length of 100 millims., the magnification is over 1000.*
In spite of this large magnification the resulting lines are fairly regular, a fact which
bears eloquent testimony to the excellence of workmanship and definiteness of

STEADINESS or B*>L*NCC UNOCR VARIOUS CONDITIONS

101-5

Cmxt'T T

Lc

.OIL*

101 0

1005

NoC

CNT.

\

\ BOTH

7 Cl

100 0

99-5

E.NT

93-0

Pf RIO >S

(«=

SECOM

09,

tMT HAN > COILS

2

5.

Fig. 20.

behaviour of the instrument. The perfection of the arrangements for reading the
scale are also of a high order, seeing that 1 millim. on the ordinates of the curves
corresponds to a length of less than 1 micron (TO^OO millim.) on the scale, and the
regularity of the curves shows that the scale can be read to an accuracy of this order
under favourable conditions.

In fig. 20 the scale is alx>ut T'o °f tnc original.

A NEW CURRENT WEIGHER, ETC. 507

From the curves it will I* seen that with no current through the coils, or with
normal current through both sets, the drift was comparatively small, Amounting in
the worst case to only 0'15 division (0'014 millim.) in five complete periods. With
current through one set only, however, the drift was much greater, amounting to
076 division in five periods in the lowest curve, the direction being such as to
indicate increase in weight of the suspended coil through which the current was
flowing. As the sensitiveness of the balance during the above tests was 0'82 division
for the reversal of 1 milligramme, the apparent rate of change of mass amounted to
0'38 milligramme per period (or 0'65 milligramme per minute), when current passed
through one set of coils only, whilst with current through both sets the greatest
change was about a fifth as great. There is, therefore, a considerable increase in
steadiness of the rest-point when both sets of coils are used.

Other advantages of two sets of coils are (a) that two independent determinations
of the ampere can be made by using the sets separately ; (b) the two sets being very
nearly alike, one serves as a check on the constancy of the other set by arranging
them in opposition and weighing the difference between their effects, which difference
should, of course, be constant for a given current ; (c) the difference in the force, if
any, produced by changing the relative positions in azimuth of the fixed and suspended
helices as suggested by Lord RAYLEIGH* can be readily found by making the
differential test above mentioned with one set of coils in a certain relative position,
whilst that of the other set is varied. The result of such a test is given on p. 517,
Section 11.

A lengthy experience with the current weigher proves that the self-checking
facilities provided in the instrument are of very great utility and form one of the
most valuable features of the balance.

SECTION 9. — FORCE BETWEEN HELICAL CURRENT AND COAXIAL CIRCULAR

CURRENT SHEET.

As mentioned in the introductory section (p. 467) of this paper, the formula used
for calculating the force between the fixed and suspended coils is due to the late
Professor J. V. JONES, viz. : —

M,) .......... (1),

the meanings of the quantities being as there defined. This formula is rigorously
exact for a helix and current sheet, and a very close approximation for two helices
of fine pitch. The order of the error is considered in Appendix B, p. 541.

The arrangement of the coils in the actual instrument may be represented
diagrammatically in section by fig. 21, which is meant to indicate a vertical section
through the vertical axes of the windings, the vertical dotted lines being the axes of

the coils.

* 'B.A. Report,' Dover, 1899, p. 292.

3 T 2

508

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

Here «j and «/ represent the lower and upper ends respectively of the left-hand
suspended coil, whilst CBD and GJH indicate the lower and upper helices on the
left-hand fixed cylinder. B and J are supposed to be on the mid-planes of the
respective helices. The right-hand suspended and fixed coils are similarly represented
by a, and a,' and C'B'D', G'J'H'.

1

H

a-',

T

e.

D

a,

r

a.

J"

—

•-• — - "~-

6-'
D'

o'

<*••

r'

, t

Fig. 21. Vertical section through coils of current weigher.

If only one pair of coils be used for making a determination, the change of apparent
mass due to a reversal of current in the fixed coils enables the current strength to be
calculated directly. Using both pairs, however, introduces cross actions between
them, and the forces due to these must be calculated or eliminated.

To distinguish the forces between the coils on one pair of cylinders from those
between the suspended coil of one pair and the fixed coils of the other pair, we have
called them " direct " and " secondary " forces respectively. For example, the forces
between alf a/ and CD, GH are called " direct forces," whilst the vertical component of
the force exerted on a,, a/ by CT)' and G'H' is called a " secondary " force. For
shortness, these are designated by D and S.

A little consideration will show that when the current in a1( a/ is in the same sense
as that in «2, a/, and both sets of coils are in use, assisting, the electromagnetic force
operative is the sum of the direct and secondary forces (D + S), whilst if the currents
in the suspended coils are opposite in sense, the resultant force is (D— S).* Two sets
of observations are therefore necessary to eliminate the secondary forces, t

Horizontal components of the cross forces will exist, as well as forces due to the
action of the suspended coils on each other tending to push them apart or pull them
together. These forces, however, are so small compared with the mass of the
suspended cylinders that no appreciable displacement is produced. Careful observa-
tion by a telescope, made with a view to detecting side displacement, led to a
negative result.

Considering one set of coils, say the left-hand ones in fig. 21, the value of (M2— Mj)

* In each case the directions of the currents in the two pairs of .coils (left and right hand) are made
such as will produce torques on the beam in the same sense.

t Mr. SEARLE has developed an expression by which these secondary forces may lie calculated.

A NEW CURRENT WKK1HKR, ETC. 509

in formula (1), p. 507, was determined as follows: — The mutual induction of one of
the two helices on CD, the lower half of the fixed cylinder, and the circle a,, was
calculated by finding M for the circle a, and helix BC (half of CD) and doubling it.
To find M, two mutual inductions were calculated, viz., that between a/ and a helix
of length JD, and that between a,' and a helix of length JC, and taking the
difference. It was therefore necessary to calculate three coefficients of mutual

v

induction ; these, for convenience of reference, are designated by M0, MS|, and M«,f
respectively. The value of (Ma— M,) for both helices* on CD is given by

M,-M1 = 2{2M.-(Me.-Mei)} ....... (2).

For the current sheet OK, a/ and the helices on GH the value of (Ms— MI) was
determined from M,, Mei, and MSi by the increment formula

'/MM '/A da , dx .

- = -+'' + t ........ 3

which gives the change in Me due to small changes in dimensions, A being the radius
of the helix, a that of the circle, x the length of the helix, and q, r, and s coefficients
determined as shown on pp. 200, 201 (Ibid.).

The sum of the two values of (Ma— M,) thus obtained gives the total for the left-
hand set of coils, and is designated by M,..

As the dimensions of the right-hand set of coils are very nearly equal to those of
the left-hand set, the increment formula was employed for finding the two values of
(Mj— M,) for this side of the current weigher and their sum called M,,. The " direct "
force between the fixed and suspended systems when arranged to assist each other

may therefore be written

.......... (4),

and the mass required to balance this force is given by

Taking the values of ML and MH determined on p. 514, and assuming g to be
981 '20, we get for both sets of coils (neglecting secondary forces)

m

0-1x184 51922-471
(for I ampere) = VI x -^^ x -^^-

= 7'49964 grammes;
or change of mass on reversal of 1 ampere = 14*99928 grammes ...... (6).

* As previously mentioned, each cylinder has double-threaded screw grooves.

t 'Roy. Soc. Proc.,' vol. 63, p. 197, 1898.

\ There are 184 turns on each suspended cylinder, the axial length of which is 12'983o centinw.

510 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:
Similarly for reversal of 1 ampere in left-hand set we get

m, = 7'49987 grammes . , . ... (7),

and for right-hand set

mr = 7 '49942 grammes (8).

Further, we may express the current in amperes in terms of the mass to halance
change of force on reversal as

Amperes = v/m/14'99928 (9),

when both sets of coils are used (secondary effects eliminated), or

Amperes = v/m/7'49987 for left-hand set . . . . (10)
and

Amperes = \/m/7 '49942 for right-hand set. . . . (11).

Again, by taking the sum of the balancing masses obtained in a D + S observation
and a D— S observation* with the same current passing, and calling this m', we have

Amperes = v/m729-99856 ........ (12),

the formula employed in the great majority of the measurements.

SECTION 10. — CALCULATION OF MUTUAL INDUCTION OF HELIX AND CIRCULAR

END OF COAXIAL CURRENT SHEET.

The formula employed is

...... (13),

where

© = angular length of helix, A = radius of helix, a = radius of circle,

x = axial length of helix,

e2 = 4 Ao/( A + a)2, c'2 = 1 - c2,

F = 4 Aa/( A + a)2 + a2, V = 1 - F,

and F, E and II are complete elliptic integrals of the 1st, 2nd and 3rd kinds
respectively ; F and E are to modulus k, and

f2 (ty

= J l-c2sin2l-Fsin2T»4

* See p. 508.

t J. V. JONES, 'Roy. Soc. Proc.,' vol. 63, p. 198, 1898.

A \EW CURRENT WEIUHER, ETC. 511

Putting c'/kf = sin ft, the quantity (F— n) can be expressed in terms of complete
and incomplete integrals of the 1st and 2nd kinds* ; thus

c-'*" sin£cosy8(F-n) = -±w-F(*) F(jf, £)+£(*) F(Jb, ft)+¥(k)E(V, ft) . (15).

The various elliptic integrals required in equations (13) and (15) were calculated
in three ways, viz. : —

(a) by interpolation from LECKNDKU'H tables ;

(b) directly by successive quadric transformation ;t

(c) directly by series. J

Method (a) was used by two of us independently, one (F. E. ti.) employing
a calculating machine, and the other (T. M.) using logs. To obtain the desired
accuracy, 1st, 2nd and 3rd differences were required in the interpolations.

As a check ou possible misprints in the tables, one of us (T. M.) calculated all the
complete integrals directly by series, and also both complete and incomplete, by
method (i). When the numerical coefficients in the series had been evaluated, the
method (c) proved quite expeditious. For the convenience of others who may not
have access to tables, these coefficients and their logs are given in Appendix A.
Successive quadric transformation, however, proved quickest when the angle ft was
well conditioned, three or four transformations being sufficient. But in the case of
Me, the angle ft was nearly 45°, and to obtain the seventh figure accurately ten-figure
logs were used.

For any particular value of M« the corresponding increment coefficients </, r, and .<?
are given by the expressions

(16).

Denoting — -J-? — * . , by Z, these may be written
-

where

* CAYLEY, ' Elliptic Functions,' § 183.

t CAYLEY, Chapter XIII.

t CAYLEY, Chapter III, § 77.

§ J. V. JONKS, « Roy. Soc. Proc.,' voL 63, pp. 200, 201.

512

PROFESSOR W. E. AYRTON, Mil. T. MATHER AND MR. F. E. SMITH:

The mean (arithmetical) dimensions chosen for calculating the values of Me, MSi,
and Me> respectively, were those of the left suspended coil and the lower helices on
the left fixed coils, and are given in Tahle VIII.

TABLE VIII.

2A = 33'00169 centims., 2a = 20'35833 centims.
From these we get

A + a = 26-68001, log (A + a) = 1-4261860,

A-a = 6-321674, log (A-a) = 0'8008325,

c2 = 0-9438574, log c2 = 1-9749064,

c = 0-9715233, log c = T'9874532,

c'a = 0-0561426, log c'2 = 27492925,

c' = 0-2369443, log c' = I '3746462.

These quantities are required for the three values of M to be calculated. The
remaining quantities differ according to the axial length of the helix taken, and are

tabulated below.

TABLK IX. — Calculation of Mutual Induction.

Quantities

Values of Quantities.

For Me.

For Me,.

For Me,.

X

6-3500

6-6328

19-3340

k2

0-8932480

0-8889178

0-6188676

k

0-9451178

0-9428244

0-7866814

k"2

0-1067520

0-1110822

0-3811324

K

0-3267292

0-3332899

0-6173590

sin B

0-7252007

0-7109253

0-3838029

cos B

0-6885375

0-7032675

0-9234148

B

46° -29' -1325

45°-18'-6152

22° -34' -167

P(4)

2-547390

2-528747

1-970492

E(*)

1-1102534
0-8198800

1-1137183
0-7991096

1-288108(1
0-3977745

E(fc'!/J)
/fc'2sin/?cos/?(F-II)

0-8029245
-0-7037136

-0-7629102

0-7826638
-0-7224003

-0-7516717

0-3901148
1-0735131

-0-4592672

^(F-ll)

F-E

it*

1-6088889

1-5918560

1-1026320

e

90*

94s-

2747T

M

5859-722
x4

6063-486
x2

11292-649
x2

23438-888

12126-972

22585-298

A NEW CURRENT WEIGHER, ETC.

513

From the above table and formula (2), p. 509, the value of M,— MI for the left
suspended cylinder and the lower helices on the left-hand fixed cylinders can be

obtained, viz. : —

= 12980-562 (18).

To determine the corresponding quantities for the remaining part of ML and for
those of MR the quantities given in Table X. are required.

TABLE X. — Calculation of Increment Coefficients q, r, and * [Equations (3) and (17)].

Values of Quantities.

Quantities.

For M«.

For Me,.

For Me,.

n

15-373275

15-165695

9-691595

IU

3-642846

S- 593424

2-296368

F-Hc-

• 1-095218

- 1-064677

-0-325876

F+ne'

6-189998

6-122171

4-266860

z

0-845979

0-840184

0-643365

PO(*)

- 0-670387

- 0-654965

-0-234772

9

• 1-04931

• 1-027088

-0-410543

r

2-25687

2-247538

2-045C30

s

- 0-20756

- 0-220449

-0-635087

Formula (3), p. 509, Section 9, may be written

e = Me -£ dA.+ Me - da+ M. - dx
A. a x

and making use of the values of M, q, r, and s, from Tables IX. and X., we get

' =- 1490-5 rfA + 51967 da-766'Qdx'}

,' = - 754-8 e£A + 2677-6 da-403'Odx 1 .... (20),

/ = - 561-9

where

....
)

M.' = 4M., M.,' = 2M^, Me,' = 2Me,,

and dA., da, and dx are increments of A, a, and x respectively.

In cases where dA., da, and dx, are the same for all the windings involved in one
value of M,— M,, the equations (20) may be combined, thus giving

d(Mt-M1) = - 1683-4 dA.+ 3335'6 da-W2dx .... (21).
For the helix GH (fig. 21) and current sheet a,a', we have

dA. = -0-00053, da = 0, dx = 0,

VOL. ccvu. — A.

3-M,) = + 1683-4 x 53 x lO'6 = 0'892.
3 u

514 PROFESSOR W. K AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

Denoting by (Ma'— M,') the new value of (Ma— M^, we get

(M/-M/) = (Ma-M,) + 0-892 = 12980-562 + 0-892

= 12981-454,
and therefore

ML = (M2-M1) + (M/-M/)

= 25962-016 (22).

For the right-hand set of coils the increments are

dA." = 0-00111, dA.'" = 0-00052,

.

for the lower and upper fixed coils respectively, and for the current sheet a2a2',

da = 0-00027.
Hence

(M/'-M/') = 12979728, (M/'-M/") = 12980726,
therefore

I MH = (M/'-M/O+CM/'-M/")

= 25960-454 (23),

and

ML + MR = 51922-47 (24).

The values obtained by the calculating machine were as follows :—

ML = 25962-04 (22'),

MR = 25960-43 (23'),

and

ML + MU = 51922-47 (24').

Thus the two methods give the same result for the sum ML + MK, although the
constituent values differ by nearly 1 in 1,000,000. It should, however, be pointed
out that one of us calculated the mutual inductions from the arithmetical mean
dimensions of the helices concerned, and the other from the calibrated mean dimen-
sions as obtained from the curves shown in figs. 13, 14, 15, and 16. The agreement
is, nevertheless, very close.

Mr. G. F. C. SEARLE has calculated the force between the current in one set of
fixed coils and that in the suspended coils of the system not coaxial with it. The
distance between the axes of the coils is a most important factor in the calculation,
the accuracy of the calculation being approximately that with which the 5th power
of this distance is known. The distance was determined as 50*8 centims. approxi-
mately, and for a current of 1*018 amperes a balancing mass of 0'04276 gramme was
calculated by Mr. SEARLE'S formula. In practice the balancing mass for this current
is G'0424 gramme. The agreement is satisfactory.

A NEW CURRENT WEIGHER, ETC. 515

SECTION 11. — DIFFERENTIAL EFFECTS OF THE SEVERAL WINDINGS, AND THEIR
RELATION TO THE LINEAR DIMENSIONS OF THE COILS.

On each fixed cylinder there are four helices, and on each suspended cylinder two
helices, and the diametral dimensions of those on the same cylinder are slightly
different. Let the upper helices on the left fixed cylinder be designated Ul and U2
respectively, the lower ones Ll and L2, and those on the coaxial suspended cylinder
a and b ; also let the helices on the cylinders to the right be represented by similar
letters characterised with a dash. Then the maximum force due to a current yk in the
left fixed helices and a current •/ per unit length in the current sheets equivalent to
the suspended helices may be written

/y*(Min. + MOT.+Mtta+Mu.+MIJU+M,w+Mlu+Mu,) = 2/yAML = DL (say),

where MU1(1 is the difference in mutual induction of the coil Ul and the circular ends
of a (i.e., Mj— M, of formula (1), p. 507), and ML is the difference in mutual induction
of all the fixed left helices and the circular ends of the current sheet equivalent to a
and 6. For the system on the right there is a similar expression which may be
denoted by DR, and the sum DL + DR is conveniently written as D.

In addition, there are secondary forces due to the mutual action of the fixed systems
and the opposite suspended ones. The maximum secondary effect due to the left
fixed system and the right suspended one may be written SL, and that due to the
other systems SH. Let SL + SK = S.

The direct and secondary forces may aid one another, in which case the total force
measured by the balance is D + S, or they may oppose one another, the force thus
becoming D-S. The sum (D + S) + (D-S) gives 2D. If only one-half of the whole
system is used, DL or DH is obtained. In the determination of the E.M.F. of the
cadmium cell, the forces D + S and D— S were measured in most cases.

Estimation of Difference between Left-hand and Right-hand Systems of Coils. — If
the two forces DL and DK act in opposition on the beam of the balance, the force
required to maintain equilibrium is (DL— DR) + (SL— SR) or (DL— DB) — (Sj,— SH). By
reversing the current through all the coils on one side of the balance, one of these
states is obtained from the other. If both of the balancing forces are measured, the
mean is DL— DR, which is equal to 2y'yA(ML— MK). Thus the mean balancing mass is
2y'yA(ML— MH)/</, and is to be accompanied by a positive sign when the force acting
on it is in the same direction as DL, and by a negative sign when in opposition to I >, .
If mi is the balancing mass, ML— MR = Wi^^y'y* ; a check is thus afforded on the
calculated difference ML— MH. The calculated value of ML is 25962 '04 centims. (see
(22'), p. 514), and of MK 25960'43 centims., the difference being 1*61 centims. The
mass ?H, was determined on five different dates, and on each occasion the current was
reversed through all the fixed coils in order to reverse the direction of the force and

3 u 2

516 PKOFESSOK W.-E. AYRTON, MR T. MATHER AND MR. F. E. SMITH:

thus secure greater accuracy. The values of 2wH obtained on these occasions are as
follows : — 0'55 and 0'48 milligramme ; 0'4 and 0'38 milligramme ; 0'5 and 0'5 milli-
gramme ; 0'36 and 0'4 milligramme ; and 0'4 and 0'3 milligramme, the sign being
such as to indicate that ML was the greater. The mean value of mj is 0'21 milligramme,
and the probable error of this value is about 10 per cent. A current of 1'02 amperes
was used, so that 2/ = 0-102x14-1724 and yh = 0'102. Hence the experimental
value of ML— MR is 0-00021^/0-147 = +1*40 centims., and is subject to a probable
error of about 10 per cent. The agreement with those independently calculated by
T. M. and F. E. S. from the dimensions of the coils (1'56 centims. and 1'61 centims.)
is remarkably good.

Estimation of the Difference in the Diameters of the Coils on the Fixed Cylinders. —
Suppose the current in Ul is in opposite direction to that in U2, those in Ll and L2
to be in opposite directions to each other, and that the currents in the suspended
coils a and 6 are co-directional. Let the system on the right be inoperative. Then
the force is

y'r* [Mula+M016-(Mu2a+MTO) ± {Mtla+ML16-(ML2a+ML26)}]5

where / and yh have the same meanings as before. Here Muio— MU2a and Mm6— MCK
are very small and practically equal; similarly MLla— ML2a is equal to MU6— M^ very
nearly. Hence the force may be written 2y'yA{(Mcla— MU2a) ± (M^-MLa,)}. By trial
this may be made a maximum. If we assume Mula to be greater than MUJa, and MUO
to be greater than ML3a, the maximum force is

2/yA{(Mma+ML10) - (M^+M^)}.

The difference of the mean diameters of the separate helices on the upper and lower
portions of the fixed cylinders was measured as 3/x for the coils on the left and as 2/u,
for those on the right (p. 488). For a mean difference in radius of 1 '2/t, the value of the
force for one system, as calculated by the last equation, is 0*02 dyne, and on reversal
of the current through the fixed coils the necessary change in the balancing mass to
maintain equilibrium should be 0'04 milligramme. If the left and right systems be
made co-operative in their effect, the change in the balancing mass will be twice this,
i.e., very nearly O'l milligramme. In experiments made to check this value, all the
possible combinations of the coils on the fixed cylinders were made, subject only to
the condition that the currents were in opposite directions in adjacent helices. Some
smadl displacement of the resting point of the balance was invariably recorded on
reversing the current in the fixed coils, but the change was exceedingly small and
not always in the same direction. The mean of the first five observations is O'O milli-
gramme as the balancing mass, and the mean of the first ten observations is O'l milli-
gramme, results which are of little value except to show that the difference in
diameter of the helices on the fixed cylinders is very small

A NEW CURRENT WEIGHER, ETC. 517

.,„

Cylinders. — When the currents in the a and 6 wires on the left suspended cylinder
are in opposition, the maximum force due to the current in the left fixed helices is

y'yk {(Mula + Mc* + MLIa+ MU.) - (MDlt+ Mt,»+ MLI4+ MU,)}.

The difference of the mean diameters of the helices on the left suspended cylinder
was measured as 3/x and the difference of those on the right as 2/i. For a mean
difference in radius of 1'2/t the value of the force for one system is O'OG dyne. If the
left and right systems are made to co-operate, the necessary change in mass to
maintain equilibrium when a current of 1*02 amperes is passed through all the coils
and reversed in the fixed coils should be 0'25 milligramme. The experimental value is
0'3s±0gl milligramme; corresponding to a difference in the mean diameters of 3'5/x± I/A.

Change of Relative Azimuth of Fixed and Suspended Cylinders. — Lord RAYLEIOH
has pointed out* that the value of the mutual induction of two coaxial helices is
dependent on the relative position of the helices, and that in strictness both helices
cannot be replaced by current sheets. The complication thence arising can be
eliminated in experimental applications by a relative rotation, since the mean field is
strictly symmetrical, and accordingly the mean mutual induction is the same as if
both helices were replaced by current sheets.

The fixed and suspended coils of the ampere balance are normally arranged, so that
the diametral plane containing the termini of the fixed coils on one cylinder is
practically coincident with that containing the termini of the coaxial suspended
coils. The mutual induction must be slightly different when these planes are at
right angles, and attempts were made to estimate this difference by experiment.
The difference of the forces exerted by the left and right systems was first determined
in the manner indicated on p. 515. One set of fixed coils was then turned through
90° and the difference again measured ; there was no certain change in the difference,
and had the change in mutual induction been as great as 5 in 1,000,000 it must have
been detected. The fixed coils of the other system were then turned through 90°
and the difference in mutual induction of the two systems again determined ; it agreed
with the previous results. The angle was altered to 60° and a few more measure-
ments made, but no change in the difference was observed. The complete set of
observations lead us to conclude that the mutual induction of the helices does not
vary with change in the orientation of the coils by more than 1 in 1,000,000.

SECTION 12. — USE OF BALANCE AND DETERMINATION OF E.M.F. OF

CADMIUM CELL.

The arrangement of the circuits employed in the determination of current strength
and of the E.M.F. of the standard cell is shown diagrammatically in fig. 22, and in
further detail in fig. 23. Fig. 24 gives a general view of the apparatus as used.

* « B.A. Report,' 1899, p. 292 (Report of Electrical Standards Committee).

518

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

The current, whose value is to be determined by the current weigher, is passed
through a standard resistance R (figs. 22, 23) and adjusted in strength until the P.D.
between the terminals of R balances the E.M.F. of the cell S. A double commutator

Fig. 22. Diagram of circuit.

C, fig. 23, with copper contacts to reduce thermal E.M.F.'s, reverses the current in
the standard resistance R and simultaneously reverses the connections to the
standard cell S. The standard resistance is described on p. 520 ; it is provided
with current and potential leads and is immersed in a tank of insulating oil. To

Slider

LJ

u

Fig. 23. Detailed diagram of circuit.

A represents an ammeter.

B
C

E
G

M

a battery.

a double commutator.

an earthed point of battery.

a galvanometer.

a main commutator.

N represents a variable resistance.
P ,, a multiple commutator and plug board.
R „ a standard resistance.
S and S' represent standard cells.
T represents a turning head for enabling either
S or S' to be used.

A NEW CURRENT WEIGHER, ETC.'

519

avoid possible electromagnetic disturbances the oil was not stirred by a motor-
driven turbine, but by a stream of air forced through it. In a few of the earlier
determinations the standard cell was kept in the room containing the remainder of
the apparatus ; considerable variations in temperature were, however, experienced,
and as there was evidence of a slight lag in the E.M.F. of the cell it was removed to

Fig. 24. General view of apparatus.

a constant-temperature room in the basement. S', fig. 23, is a second standard cell
for the preliminary adjustment of the current, and the turning head T readily allows
of either cell being inserted in the potentiometer circuit. B is a battery of 55 accumu-
lators of 30 ampere hours' capacity; it is earthed at one point to eliminate electrostatic
effects (see p. 525). The resistance of the circuit can be adjusted by means of a set
of manganin coils and a mercury trough in N ; in all there are ten 10-ohm coils,

520 PROFESSOR W. E. AYKTON, MR T. MATHER AND MR. F. E. SMITH :

tea 1-ohm coils, and ten 0'1-ohm coils in series with a mercury trough of resistance
0'12 ohm ; a sliding short-circuiting contact provides the final adjustment, a move-
ment of 3 millims. of the slider corresponding to a change in current of 1 in
100,000. The manganin coils are- wound on long brass tubes and are immersed
in paraffin oil, the capacity of the tank being 6 gallons ; a very constant
current was in this way ensured. Under favourable conditions, i.e., when manganin
formed by far the greater part of the resistance of the circuit, a current constant to
2 in 1,000,000 could be maintained for an hour or more ; when the coils of the
balance were in circuit a current steady to about the same limit could be held for
a few minutes only. This, however, is all that was desired.

The potential circuit included the resistance coil R, the cadmium cell S, a contact
key, and a galvanometer G. The galvanometer was of the Broca type, having
a resistance of 1000 ohms. The controlling field was varied from time to time and
hence the sensitiveness was not the same in all of the determinations ; in general
a deflection of 5 millims. on the scale (T5 metres distant) corresponded to a change of
one-hundred-thousandth of an ampere in the main current. The galvanometer,
commutators and all of the auxiliary apparatus belonging to the balance were made
by Mr. MURFITT, the instrument maker attached to the National Physical Laboratory.
Much of the fitting was also very ably done by Mr. MURFITT.

The Resistance Coil B, fig. 23, used as a secondary standard (numbered L. 87), is
made of thick manganin strip, wound non-inductively on six posts and insulated
therefrom by silk ribbon and shellac. The coil was built and annealed by Mr. MELSOM
in July, 1905, and its resistance changed very rapidly for many months afterwards ; it
is still rising in value. It is provided with potential points and can carry a current of
10 amperes without abnormal heating. In July, 1905, the coil was directly compared
with the mercury standards of the National Physical Laboratory, and again in March,
1906 ; the intermediate and subsequent evaluations were made by comparing it with
standard coils. The methods of comparison are described elsewhere.* The tempera-
ture coefficient was determined in 1905 and again in March, 1907 ; the mean
coefficient for the range 10°C.-20°C. is +0'0019 per cent, per 1°C., but for the
reduction of values to a common temperature a resistance chart was used. Owing
to the rapid rise in resistance with time the coil was compared with practically
constant standard manganin coils on each day a determination of current was made ;
the secular change in resistance was thus eliminated as a source of error in the
comparison of results.

The Main Commutator (M, fig. 23) is formed of four brass quadrants of square
section and an ebonite turning head carrying two springy copper contact pieces
insulated from each other. Connection with the concentric cable is made by
drilling two opposite quadrants, one aperture being £ inch in diameter and the

* "Methods of High Precision for the Comparison of Resistances," F. E. SMITH, 'B.A. Report,'
Section A, 1906.

A XKW (;ri;i;I.M \VKI«;ilKI!. KTC.

other i inch. Thin brass tubes, projecting outwards for A inch, are tilted into
thi-.se holes and hard soldered to the quadrants. To the larger of these tubes the
strands comprising the outer conductor of the concentric are soldered, and to the
smaller the inner strands are similarly joined. Each quadrant i« drilled centrally

FIXED LEFT
U2 LI L2

FIXED RIGHT
LI U2

Fit;. 25. Multiple commutator HIU! plug lioard.

with a large tapering hole; by means of conical plugs connection with other circuits
— for the measurement of insulation resistance, &c. — can thus be made. The two
positions of this commutator are designated hereafter by the letters A and B.

Multiple Commutator and Plug Jioattl (P, fig. 23 and fig. 25). — This consists of
four commutators, constructed in n similar manner to that already described, and a

Vol.. O VII. A. 3 X

522 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

plug board divided into two sections for the left-hand and right-hand sets of coils
respectively. A commutator allows of the reversal of the current in all the coils on
any one cylinder, and the plug board allows of the reversal of the current in any one
or more helices on the fixed cylinders. The inner and outer conductors of the
concentric cables, D, fig. 25, are soldered to thin brass tubes let into brass blocks and
pass to the various coils of the balance. Each helix is designated by a word, a letter,
and a number, which are marked on an ebonite bridge at the top of fig. 25 ; the
turning heads are also marked so as to enable reversals of the current to be quickly
made without likelihood of error. Each plug hole is numbered, and a scheme was
drawn up so that any desired combination was represented by a series of numbers
for the plugs and by letters for the commutators. For example, in taking a (D + S)
observation, plugs are inserted in the holes 13, 2, 3, 16, 5, 18, 19, 8, 21, 10, 11, 24,
and the four commutators arranged in positions represented by the diagram a, fig. 25.
Here the straight lines in the circle represent the directions of the turning heads
of the commutators. When the main commutator M, fig. 23, is in the A position,
and the multiple commutators as shown in a, fig. 25, the whole arrangement is
designated by the symbols «A. Reversing the main commutator changes it to aB.
Turning the commutators connected with the suspended cylinders to the positions
indicated by diagram b, fig. 25, we get an arrangement symbolised by 6B, and a
reversal of the main commutator gives 6A. Changing from «A to 6B reverses
the current in the fixed coils only, as also does the change 6B to e*A. In fig. 25 the
letters DD, &c., indicate the ten concentric cables running from the plug board to
the balance.

The reversal of the current in one of the two helices on a suspended cylinder is
made by a small commutator on the three-limbed spider. This has been described
on p. 500, and is illustrated in fig. 17.

Balancing Masses. — The weights employed are eight in number and are made of
aluminium. They are divided into two sets: four for the (D + S) observations and
four for the (D— S) observations, and the masses of the weights in each set are
approximately equal. The force due to one (D + S) weight very nearly balances half
the total force due to the current in such observations, and it may be employed for
observations of the maximum force when the current flows through all the coils of
one system and through the suspended coils only of the other system. Similarly, a
(D— S) weight may be used for observations of the minimum force under such
conditions. For the (D + S) observations two weights are used on each side of the
balance, similarly for the (D— S) observations, and the total mass of the eight weights
is required for the calculationa The masses of different combinations were, however,
also determined. The standard mass employed was a 100-gramme weight standardised
at Sevres, and the effective mass of the eight weights in a medium of density O'OOl 196
was determined as 31 '12494 grammes, the four (D + S) weights being 1573135
grammes and the four (D— S) weights being 15 '39359 grammes. Aluminium is not

A NEW CURRKNT WEIGHER, KTC. 523

a very desirable material for weights, o \vini; to its density being so small, but in
our experiments the effective mass <>f the weights never varied by so much as
8 in 1,000,000 from the mean, and if no correction had been made for variable air
displacement, the error in the measurement of the current would never have exceeded

4 in 1,000,000. Of course, the corrections were applied. The probable error of the
effective mass is of the order 1 in 1,000,000.

Preliminary Difficulties.

(A) Defects in flexible Concentric Cable. — The cable originally used consisted of
an inner conductor of 30 copper wires of diameter 0'0048 inch and an outer tubular
conductor of 74 strands of the same diameter. After connecting the balance coils
to the multiple commutator and plug board the cable was found to be faulty, and
subsequent examination showed that many of the internal strands were broken.
The cable was therefore replaced by a concentric one having an inner conductor of
three copper wires of diameter 0'022 inch, and an outer tubular conductor of sixteen,
of diameter 0-0148 inch. This proved to be entirely satisfactory.

(B) Unsteady Current and Convection Currents of Air Produced by the Heating
of the Flexible Leada. — Originally the current was led into and out of each pair of
suspended coils by two silver-gilt strips, each 13 centima long, 0'37 inillim. broad,
0'035 millim. deep, and of 0'15 ohm resistance. In each pair of silver strips there
was 1 calorie of heat produced every 14 seconds when a current of 1 ampere passed
through them, and the maximum increase in temperature of the strips was about 20° C.
The temperature coefficient of electric resistance of silver is 0*36 per cent., hence the
increase in resistance of the four strips was 0'04 ohm, and a fluctuation in temperature
of 1° C. corresponded to a change in resistance of the circuit (110 ohms) of 0'002 ohm.
Such a change in temperature frequently resulted, as was proved by including the
silver strips in a circuit containing 110 ohms of manganin ; with a current of
1 ampere the fluctuations in current were of tmT order 5 in 1,000,000. When the
strips were removed from the circuit, the corresponding changes were 1 in 1,000,000.

The energy of motion of the air particles in the immediate neighbourhood of each
pair of strips was increased at the rate of about 3x 10* ergs per second. The effect
of the convection currents of air thus produced was tested by passing a current of
1 ampere through one pair of the strips inside the balance case when the balance
coils were not included in the circuit. After the circuit had been completed for

5 minutes the resting point of the balance changed by an amount equivalent to an
added load of 9 milligrammes on that side of the balance with the heated strips ;
after 10 minutes the change corresponded to 24 milligrammes ; 15 minutes afterwards
to 39 milligrammes, after which the resting point of the balance was approximately
constant. The circuit was broken for 1 5 seconds and the change noted ; it corre-
sponded to 0'4 milligramme; equilibrium was restored after 5 minutes. The length

a x '.'

524

PROFESSOR W. K. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

of one arm of the balance is 25'4 centims., and the " equivalent arm " of each pair of
strips is about 15 centims. ; hence the maximum downward force on the strips was
65 dynes, equal to that produced by 66 milligrammes. It is interesting to note that
the mass of the two strips was less than this — being only 36 milligrammes. The
elasticity of silver changes with temperature and the control exercised by the strips
must in consequence have varied with it ; calculation shows, however, that the effect
was negligible.

To remedy these defects, each of the four strips was replaced by 80 silver wires
1 mil (0'0025 centim.) in diameter. The surface per centimetre length of the strip
was 0'80 sq. millim. and the section of the strip 0'013 sq. millim. ; the corresponding
values for the 80 strands are 6'0 sq. millims. and 0'04 sq. millim. The length of each
strand is 10'5 ceutims., and the resistance of the 80 is about 0'037 ohm ; the heating
effect is, therefore, one-quarter of that formerly experienced, and the radiating
surface over seven times as great. The sensitiveness of the balance is greater than
when the strips were used, and the current through the fine wires can be kept very
constant. In addition there is no drift in the resting point of the balance due to
convection currents of air rising from the silver wires. Fusion of the silver did not
result when a current of 07 ampere was passed through one strand.

(C) Heating Effect of Current in. Balance Coils. — The total resistance of the fixed
and suspended coils is 71 ohms at 17° C. With a current of 1 ampere the heating
effect is considerable and the resistance of the coils changes comparatively rapidly.
The following table (XI.) gives the resistance of the balance coils and estimated
temperatures when currents of 070 and 1*02 amperes respectively pass through the
coils until the latter are in a steady thermal state. In each case the circuit was
completed for 24 hours.

TABLE XL

No current.

0 • 7 ampere.

1 • 02 amperes.

Coils.

Resistance

Tempeni-

Resistance

Tempera-

Resistance

Tempera-

in ohms.

ture.

in ohms.

ture.

in ohms.

ture.

° £.

"C.

0.

On left fixed cylinder . .

26-641

17-36

27-506

25-4

28-103

:!0-4

„ right „ „ . .

26-647 17-35

27-528

25-4

28-120

30-4

„ left suspended cylinder

N8-710

17-35

9-077

27-4

9-354

34-5

,, right „

8-730

17-35

9-110

27-7

9 • 375

34-5

Temperature of balance case

—

17-35

— —

82-0

—

24-2

Afterwards the balance case was covered with blankets and similar observations
made with a current of 1 ampere. The maximum increase in temperature was
22° C., the temperature of the air within the balance case being 12° C. lower than
that of the suspended coils.

A NEW CURRENT \VKIGHER, ETC. 525

An idea of the effect of the convection currents of air rising from the fixed and
suspended coils was obtained from observations on the balance pointer when the
forces acting on the suspended systems were in opposition. In such a case small
variations in current strength have no measurable effect on the total force. With the
Ixilance case covered with blankets and practically uniform radiation in all directions
(the observations were made at midnight), the mean doubled rest- point of the balance
pointer was deduced from 108 readings as 20(5 '7. These readings were taken in three
sets. The first set of 36 readings gave 206 '2 as the rest-point ; the second set were
taken immediately after the first and gave 205*9 ; there was an interval of half an hour
between the second and third sets, the mean of the latter being 208 '0. The average
ditfrieiice l>etween the first 36 readings and 206'2 is 0'8, so that the mean of a few
iv.ulings is associated with a large probable error. In addition there was difficulty in
maintaining a very steady current through the heated coils ; the rest-point of the
balance was subject to drift ; and the difference of temperature between the coils and
marble and between fixed and suspended coils introduces serious difficulties in the
calculation of the mutual induction.

The rest-point of the balance is very constant when no current is flowing through
the coils and lias not passed for some hours previously ; it is also very constant for
the first 20 minutes after the circuit has been completed. The resistance of the coils
increases considerably in this period, but observations proved that a current constant
to 2 iii 1,000,000 and often to I in 1,000,000 could be maintained for four minutes, in
which interval the resistance of the balance coils increases about 0'12 ohm, and the
sliding contact of the mercury trough passes from the most to the least favourable
position for adjustment. In this interval three readings of the balance pointer could
always be taken, and experience has shown such readings to be remarkably accurate.
This method was adopted.

(D) Electrostatic Effect*. — Electrostatic effects of sufficient magnitude to produce
a readable deflection of the balance }>ointer were not antici]>ated. The mean electro-
static potentials of the various pairs of coils are, of course, different, hut the maximum
variation between any part of one suspended and any part of one fixed system is less
than 36 volts when a current of 1 ampere is flowing. A test was made by connecting
the upper coils of one fixed system and the lower coils of the other fixed system to
the + pole of a lottery of 110 volts; the other coils of the balance were connected
to the — pole and to earth. No difference was observed in the rest-point of the
Ki lance, and hence there could be no disturbing effect due to electrostatic attraction
l»tween the fixed and suspended coils. When, however, the balance coils were placed
in series and a current of 1 amj)ere passed through them, a difference in the rest-point
of 07 scale-division was always observed on reversing the current ; this was found to
be a measure of the difference of the electrostatic forces between the suspended coils and
the metal guard-discs d, fig. 6 (Plate 8), underneath them. The difference of mean
potential of the coils on the suspended systems is 62 volts ; the metal rings are about

.r>2<; PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH

3 millims. distant from the bases of the cylinders and are practically earth-connected.
If the mean potentials of the suspended systems were +31 volts and —31 volts
respectively, relatively to the earth, then the total electrostatic effect should be nil.
The resistance of the whole circuit was 110 ohms, that of the balance 71 ohms, and
the E.M.F. of the battery was 110 volts. By earthing the battery between the 19th
and 20th cells, counting from that end directly connected to the balance coils, the
potentials were approximately as indicated and there was no measurable effect on
the balance when the current was reversed. Except for the first few observations
the battery was earthed at this point No error was introduced by the omission, as
the electrostatic effect occurs twice with opposite signs in the observations.

Normal Procedure. — In making a determination of the strength of a current, the
following scheme was adopted : —

(1) The commutators and plugs were set so that the current circulated through
the balance coils in the order : suspended left, fixed left, fixed right, and suspended
right, and so that the total force was the sum of the direct and secondary forces
(D + S) (see p. 508). Observations for the determination of the balancing mass were
then made and repeated when

(2) the current through the fixed coils was reversed ;

(3) the current through the whole of the balance circuit as typified by (2) was
reversed ;

(4) the current through the fixed coils was reversed, that in the suspended coils
being as in (3) ;

(5) the current through the whole of the balance circuit as typified by (4) was
reversed.

Each of these arrangements as indicated by two letters, one denoting the position
of the main commutator M, fig. 23, and the other that of the commutators on the
multiple commutator and plug board P, as described on p. 522. After these obser-
vations a similar set was made when the direct and secondary forces opposed one
another, thus determining (D— S). The order of making the observations in each set
was rigidly adhered to, but the (D— S) observations sometimes preceded and sometimes
followed the (D + S) observations.

After the first few determinations of E.M.F. had been made, the current which it
would be necessary to pass through the circuit to balance the cadmium cell was
estimated from the secular change in resistance and the temperatures of the coil
and cell ; the balancing mass was then calculated and the position of the rider
decided on, so as to give, together with the weights, the required mass. Previous to
observations of any kind being made, the circuit through the manganin coils was
completed for an hour or more, after which an examination of the steadiness of the
current was made by one of us, and observations of the sensitiveness of the balance
and stability of the rest-point of same were made by another. In accordance with
the scheme on p. 522, the multiple commutator was appropriately set, the balancing

A NEW CURRENT WKKUIKK, ETC. 527

weights placed in position, and, at a given signal, the balance coils were included in
the circuit. The resistance in N, fig. 23, was rapidly adjusted until (1) the ammeter
reading appeared to he the same as hefore, (2) balance waa obtained when S' was in
the potentiometer circuit, and (3) the fulfilment of the latter condition when 8 wan
substituted for S'. In general, these adjustments occupied about 10 seconds. When
• '"in li tii u i (2) held, a signal was made to the balance operator, and the beam of the
balance was freed. The average duration of a complete set of observations was
-iJ minutes, and during this time the balance coils were included in the circuit for
about 12 minutes.

General Behaviour of the Brilance. — After eliminating the difficulties mentioned
on pp. 523-520, the working of the balance, when cold, was most satisfactory. Under
normal conditions the constancy of the rest-point of the balance is well within O'l
scale division when no current passes through the coils, and the sensitiveness is about
8 divisions for 10 milligrammes. When a current, passes through the coils for not
more than 20 minutes the same constancy is in general maintained, and if the
balance circuit is occasionally broken — as it is in experiments for the determination
of current — this interval of constancy is prolonged to 30 minutes or more. If the
current through the balance coils is maintained after this interval, approximating to
30 minutes, the balance Incomes unsteady, and no very accurate observations can be
made ; if, however, the circuit is broken after the interval, the balance reading remains
approximately constant, variations of the order of 0'2 scale division only being
observed. At the end of three or four hours another determination of current is
possible, with practically the same degree of accuracy as before, but soon after these
observations the balance becomes unsteady, and shows variations in the rest-point,
gradually increasing from O'l to I'D scale division. If the second set of observations
are made within one or two hours of the first set, the balance reading is not constant,
and the results obtained are not of a high order of accuracy. In general, therefore,
only two determinations of current are possible within six hours, but these are
associated with a very small observational error. One determination normally
occupies from 16 to 25 minutes.

Our usual procedure was to make one complete set of observations in the morning
and another in the afternoon, after the balance had been cooling for several hours.
Attempts made on several days to make a third set were never successful.

The time which elapsed tat ween morning and afternoon observations of E.M.F.
was usually devoted to silver-deposit determinations, the standard cell S and
resistance R, fig. 23, l>eing used for keeping the current steady at a calculable value
during the deposition. In effect, therefore, the combination of cell and coil, forming
a secondary standard of current, was standardised morning and afternoon by the
iNilance, and used in the interval for measuring the current through the voltameters.
As, however, the determination of the electro-chemical equivalent of silver forms the
subject of another paper, it need not be discussed here.

528

PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

Below is a sample series of readings taken in the second determination on .Jan. 2,

1906:—

January 2, 1906.
(1) Observations for constancy of resting point and of sensitiveness: —

Constancy : —

99-5 100-85T
99-5 100-8 ^
99-4^,

200-3,,

1-6, 100-6 I
1-7 100-6 }• 2
1-7 J

The second set of observations
was made 10 minutes after
the first set.

Sensitiveness : —

10 milligrammes
on left.

100 -7r, 107 •;

100-8 107-2 ^ 208-0,

100-86

10 milligrammes
on right.

90-5 102 -061
90-6, 102-0 > 192-6
101 • 9, J

Sensitivenes8 = 0-77 division for
1 milligramme.

Cell. Coil. Case. Barometer.

°C. °C. °C. centiins.

Temperatures at commencement of observations . . . 16-71 16-70 14-0 74-5

(2) Determination of current : —

>mmence
conclusion

16-71

16-78

14 -L>

(D-S).
Position of rider, -
Commutator

0-0060.
Weights

Position
Commutator

(D + S).
of rider, -0-0050.
Weights

positions.

on

positions.

on

rtA

99
99

•85
•8S

101-0,

}

200-9

I'

aA 98-0
98-0

101

'6 j 199-6

R

bB

98

•7

101 -95

"I

200-67

R

fcB 99-9*

100

'7 1

I

98

"7,

J

99-9,

J

bA

98
98

•4,

•4,

102-1

}

200-5,

R

. 4A 99-9,

100
100

•6 \

•5, J

L

aB

100
100

•0

•o

100-9,

}

200-9,

L

aB 99 • 65
99-6,

100

'°6 | 199-7

R

aA

99
99

•1
•1

101-7,

I

200-8,

L

aA 99-2
99-2

100

'4(i | 199-6,

R

Mean "a" reading = 200 • 90 \ difference
„ "b" „ = 200-6,/ = 0-2,

Mean "a" reading = 199 -6, \differerice
„ "b" ' „ = 200-59/ =0-94

Effective mass of
weights

Balancing mass

= 15 -3935,, -0-0000:,

= 15 -39354- 0-0120

-0-0004
= 15-38114

Effective mass of
weights

15-7313, -0-0000,

Sum of balancing masses •-
Mean current = s/31- l6T2/ N/4 x 7

Ilakncing mass = 15-73130 -0-0100

-0-0012
= 15-7201,,

= 31-1012 grammes.
49964* = 1-018212 in amperes.

* Formula (12), p. r>10.

A NEW CURRENT WEIGHER, KT< .

529

The "a" and "6" positions refer to the multiple commutator and plug board,
fig. 25, and the "A" and "B" positions to the main commutator M, fig. 23, as explained
on p. 522. A change from "a" to " b " reverses the current in the suspended coils,
and a change from "A" to " B" reverses the current in all the coils of the balance.
Centigramme riders were employed, and the position — O'OOGO in the (D— S) experi-
ment indicates that the rider and balancing weights were on opposite sides of the
beam. The correction — 0'00006 gramme is for the difference in density of the air
from 0 '00 1 196. In all cases the sum of the balancing masses was computed to
O'l milligramme.

TABLES OF RESULTS.

The following tables, XII. and XIII., give particulars of determinations from
September, 1905, to April, 1907 ; no determination has been omitted, except when

TABLE XII. — (Cadmium Cell No. 2.) E.M.F. Determinations, using One Set

of Coils.

Date.

Obser-
vation.

Bnlanciug
mass
in

Mean temperature —

•
value of
resistance
coil in

C-

value
of

CxR.

CxR

corrected
to

Of

grammes.

Of
cell.

resistance
national

current.

17° C.

cou.

ohms.

12.9.1905

D

7-7775

16-1

1

16-9 0-99988,

1-01834

1-01822

1-01819

14.10

„

7-7833

9-51

10-35 74,

72

47

21

14.10

30

9-71

10-69

75T

70

46

19

23.10

36

9-0

8-9

73s

740

470

19.

23.10

12 11-64

11-45

79« 58,

37S

16g

24.10

23 11-80

9-55

748

65s

89|

19,

24.10

24

10-50

11-29

79,

66,

45o

21,

Mean . .

1 -01819s

30.9.1905

DB

7-7797

13-8

14-75

0-99984,

1-01852

1-01836

1-01822

14.10

7-7836 9-63 10-53

75.

77

52

26

23.10 „ 33 9-25 9'0

737

75,

48.

21.

23.10

11 11-60 11-15

790

60;

39,

18,

24.10

20 11-80 9-72

75,

66,

41.

21,

24.10

20 10-31 11-12

788

66«

45.

20,

Mean . .

1-01821,

Balancing masses for reversal of 1 ampere : —

Left-hand coils mj = 7 • 49987 «.

Right-hand „ mr = 1 • 49942

Mean of both \
seta*- J

1- 01820s

VOL. CCVII. — A.

» Formula (7) and (8), p. 510.
3 Y

530 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

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A NEW CURRENT WEIGHER, ETC.

531

3 Y 2

532 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

the observations were of such a nature that a decision to disregard the result was
arrived at before its computation. Such occasions were very rare.

Table XII. gives the results obtained when only one set of coils (left or right) was
made use of, so that there were no secondary forces to be eliminated. They are
inserted to show the order of the agreement attainable in this way, and are not
considered to be so reliable as the values deduced from the (D + S) and (D— S) tests.

When both sets of coils are operative, the balancing mass for 1 ampere is 7 '49964
grammes — this has been denoted by m in Table XIII.

PROBABLE ERRORS.

The mean error of a single observation in Table XIIL, viz., 6 parts in 1,000,000, is
surprisingly small, for this comprises the error of the balance reading, the inaccuracy
of the estimation of the secular change of the secondary standard resistance coil, the
variation in E.M.F. of the standard cell (including polarisation during the observations),
uncertainty in temperature readings, and the error introduced by the non-main-
tenance of an absolutely steady current. The probable observational error of the
mean value of CxR at 17° C. is less than 1 in 1,000,000.

The probable error of the ratio of the diametral dimensions of the coils, viz., 5 in
1,000,000, and the uncertainty in the axial dimensions of 15 in 1,000,000 (intro-
ducing a possible uncertainty in the value of the mutual induction of about 5 in
1,000,000 and in the measurement of current of about 1 part in 100,000) have not
been under-estimated. Evidence in favour of a small error is afforded by the satis-
factory agreement of the calculated and observed differences of the forces due to the
left and right systems when a current of 1 ampere circulates through them ; in
addition there is the estimate of the difference in radii of neighbouring coils from
observations of the force (p. 516). These measurements lead one to suppose that the
errors have been closely approximated to. The electrical method of setting the coils
in position has been shown to be subject to an error not greater than 1 in 5,000,OoO
of the mutual induction ; the magnetic susceptibility of the parts of the balance and
its support is negligibly small, and the effect of the current in the leads to and from
the suspended systems is too small to be measurable. The magnitudes of the errors
arising from the finite thickness of the wire used, and the assumption that one of the
coils is a current sheet instead of a helix, are discussed in Appendix B, and shown to
be practically negligible.

The possibility of error due to the oscillation of the suspended systems has not yet
been considered. For a small axial displacement of the suspended coils the force is
(1 — 11 x 10~8c?2) times the maximum,* where d is the axial displacement in mils from
the plane of minimum mutual induction. One division of the pointer scale is equal
to 3 '75 mils (= 95/j.) ; the length of the pointer is 14' 6 inches (37 centims.) and half

* See p. 502.

A NEW CURRENT WEIGHER, ETC. 533

the length of the beam is 10 inches (25'4 centime.). A difference in doubled rest-
point readings of 7 '9 divisions corresponds therefore to a difference in the mean axial
positions of the suspended cylinders of 10 mils (254/ii). In an experiment intended
as a check on the expression obtained for the change of force with small axial
displacements, the doubled rest-point in a (D + S) experiment was 197*7 in one can
and 189 '3 in another, the latter reading being obtained by loading one end of the
beam with 10 milligrammea The correct position for maximum force corresponded
to a pointer reading of 200'0. In the two experiments the difference in the balancing
masses was 0-3 milligramme, corresjxHiding to a difference in force of O'OOl, per cent.,
and from the readings of the doubled rest-points a difference in force of O'OOl, per
cent, is deduced. The agreement is satisfactory. It follows that in a determination
of current strength the doubled rest-point must not differ from the reading corre-
sponding to the position of maximum force by more than 8 divisions if the error
introduced by the difference in the positions is to be less than 5 in 1,000,000. When
making the observations, the results of which are tabulated in Table XIII., the mean
displacement of the suspended coils was always kept within 2 divisions by adjusting
the position of one of the riders, and the mean displacement for all the observations
is 0'5 division. The greatest error introduced on any occasion was therefore about
1 part in 3,000,000.

We have also to consider the relation between the amplitude of swing and the
effective force due to the current. This relation was determined experimentally. In
a particular D + S experiment the amplitude of swing was varied from 1 division to
28 divisions, but the estimated forces were identical. Other observations confirmed
this result, and it was only when the amplitude was very large and the errors of
observation great that any difference was observed ; even these differences are of
opposite signs and point to the forces being identical. It is certain that within the
limits 0 to 28 divisions for the amplitude there is no measurable difference in the
effective force. When determining the value of a current the amplitude was in
general from 3 to 4 divisions ; there is therefore no correction to be applied for the
amplitude of swing.

The remaining source of error is due to an uncertainty in the value of gravity.
No absolute determinations of g have been made at Teddington, and it was
necessary to compare the values at Kew and Teddington by pendulum observations.
Mr. E. G. CONSTABLE, of the Observatory Department of the National Physical
Laboratory, made such observations in March, April, and July, 1905. The pendulums
swung were half-seconds pendulums, the property of the Board of Education and
used in the " Discovery " Antarctic Expedition. At Teddington two positions were
chosen : one was on the concrete block on which the ampere balance stands, and the
other was in a lower room maintained at a very constant temperature. At Kew the
pendulums were swung in the north room of the small house to the west of the main
building. The difference in period of the half-seconds pendulums was determined

534 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

to be 26xlO~7 second, the period at Teddington being the greater. Excluding
observations made over 30 years ago, only two comparisons have been made
interconnecting Kew with a station where g is believed to be known in absolute
measure. The first of these comparisons was made by VON STERNECK in 1893, and
the second by Mr. G. R PUTNAM (U.S. Coast and Geodetic Survey) in 1900. The
former of these observers assigned the value 981 '160 to Kew and 981 '200 to Greenwich ;
Mr. PUTNAM'S values are 98T199 and 98T187 respectively. It will be observed that
VON STERNECK makes the value at Kew less than that at Greenwich, but all other
observers make it greater.* Also the differences found between Kew and Greenwich
by the latest and most complete observations (those by PUTNAM and by BURRAND,
CONSTABLE and LENOX-CONYNGHAM) are very close to that given by theory.
VON STERNECK observed on only two days at Kew as against six at Greenwich ; thus
the probabilities of serious error are much greater for Kew than for Greenwich.
VON STERNECK'S value for Greenwich exceeds PUTNAM'S by 0*013, but this, if we may
judge from the difference 0'019 between their values for Potsdam, represents largely
a difference in what answers to their base values. HELMERT has accepted for Kew
the value 981 '200, t and it appears that no serious error is introduced by our
acceptance of this value. From Mr. CONSTABLE'S observations the value of g at the
National Physical Laboratory would therefore appear to be 981"19 centims./sec2.

The theoretical difference between Kew and Teddiugtou may be obtained from
VON HELMERT'S formula. The places are very similarly situated with respect to
surface strata and surroundings, and the only corrections it is necessary to apply are
those for difference of latitude and difference of level. The latitude of Kew is
51° 28' 6", and of the National Physical Laboratory 51° 25' 20" approximately;
the level of PUTNAM'S observations at Kew was 17 feet above mean sea-level, and at
Teddington the mean level of -Mr. CONSTABLE'S observations was about 34 feet. The
correction for difference of latitude is — 0'0044, and for the difference of level it is
— 0-0010 ; the theoretical value is therefore 981'196 if Kew is 981'200. The probable
error of any accepted value depends, of course, on the errors of the intercomparisons
and on the error of the absolute determination at the base station. It appears that
these are not very large, and that we may accept the value 981'19 ceutims./sec2 as
correct to 3 in 100,000.

The determination of current by means of the ampere balance is therefore subject
to errors of the following magnitude : —

(1) Due to uncertainty of dimensions of coils : possible error about ±0'001 per cent.

(2) Due to uncertainty in the value of g : possible error about ±0'0015 per cent.
All the other sources of error introduce uncertainties less than ±0'001 per cent.,

and may be disregarded. The total error of an estimation is therefore of the order
±0-002 per cent., or 2 in 100,000.

* See G. P. LENOX-CONYNGHAM, ' Roy. Soc. Proc.,' A, vol. 78, p. 246, 1906.
t 'Report, Geodetic Conference of 1900,' p. 321.

A NEW CURRENT WEIGHER, ETC. 535

As numerous determinations of the balancing masses for (D + S) and (I)— S) have
l>een made, the value of S for 1 ampere can be calculated from them with considerable
accuracy. By using this value a determination of current, using both sets of coils,
can be made by taking the apparent change of mass produced by a single reversal of
current in the fixed coila The necessary observations can lie made in less than five
minutes, so that a very short time would suffice for making an absolute determination
of current in this way.

History of the Standard Cell employed. — When the first determination of current
was made, the cadmium cell chosen for insertion in the potentiometer circuit was one
whose E.M.F. was lower than that of normal cells by O'll millivolt. Originally it
was not proposed to use this cell permanently, but as its previous history indicated it
to have remained very constant, it was afterwards decided to do so. The cell was
compared with other standard cells on each day that a determination of current was
made and on many other intermediate days. All the cells were constructed in the
manner descritad by one of us (F. E. S.) in the 'Report of the British Association,'
Section A, 1905, and were set up at the National Physical Laboratory. In the first few
determinations the cadmium cell was in the same room as the ampere balance, and its
temperature sometimes varied from 6°C. to 19°C. within 24 hours. Careful observa-
tions showed that the E.M.F. of the cell did not very closely follow this rapid change
in temperature, and the corrections to the value of CxR in Table XIII., Column 9,
were obtained from a curve which, though not very different from the temperature-
coefficient curve of the cell, is not identical with it. This statement applies to the
first twelve observations only, for on and after November 23, 1905, the cell was kept
in the resistance-standards room, which is maintained at a nearly constant tempera-
ture of 17°C. After November 23, the correction to 17°C. was obtained from the
temperature-coefficient formula

E, = E17-3-46xlO-6(*-17)-0-OGGxlO-8(<-17)1.

•

This formula is the result of a determination made at the National Physical
Laboratory, the range of temperature during the observations being 10°C. to 30° C.
The coefficients are practically identical with those given by JAEGER and KAHLE.*
Their formula is

E( = 1 '0186-0-000038 (<-20)-0'00000065 (<-20)J.

The cell employed in the potentiometer circuit (hereafter called No. 2) was set up
in January, 1905 ; those with which it has been compared were set up on various
dates ranging from October, 1904, to April, 1907. The comparisons indicate that the
cells have remained constant within a few hundred-thousandths of a volt, or have
changed uniformly. The actual differences between the cells are not given here, but
may be summarised by saying that with the exception of cell No. 2 the greatest

* 'Zeitechr. f. Instranentenk.,1 1898, p. 161.

536 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR F. E. SMITH:

difference in E.M.F. of any cell from the mean E.M.F. of all of them is 0'03 millivolt,
and the difference between the mean E.M.F. of the old cells and the new cells set up
in March and April, 1907, is 0'02 millivolt. This comparison indicates constancy of
the old cells ; Table XIV. confirms this view. The mean difference of the " old and
new cells" and cell No. 2 is O'll millivolt. On September 13, 1906, and on April 10
and 11, 1907, a cell representing the mean normal cell was employed in the deter-
mination of current. The results are contained in Table XIV.

The mean value of CxR at 17° C. is 1'01830»; the value from comparison with cell
No. 2 is 1'018307. Both these values assume g to be 981 '20; correcting for the
difference of this and the accepted value 98T19, we obtain

' T01830 " "":.

as the mean value of CxR at 17° C.

It should be pointed out that the " international ohm " used in these measure-
ments is that employed at the National Physical Laboratory, which unit does
not differ by more than 3 parts in 100,000 from that of the Reichsanstalt. In
absolute measure, however, its value is not known to a high degree of accuracy.
Taking its ratio to the Board of Trade ohm as determined by one of us (F. E. S.) in
1903 ('B.A. Report,' 1903, and 'Phil. Trans.,' A, voL 204) as I international
ohm = r0001B B.O.T. ohm, and assuming that the B.O.T. unit has remained constant
since 1897, when its value in C.G.S. units was found to be 1 '00026* x 109, we get
1 international ohm = r00041xlO* C.G.S. units, and the E.M.F. of the normal
cadmium cell at 17°C. becomes

\

T0187, x 108 C.G.S. units (approximately).

This number must, however, be considered as provisional only, pending a re-
determination of the international ohm in absolute measure.

It is of interdst to compare our value of C x R in terms of the international ohm
with that obtained by GUTHE in 1906.t He gives the number 1 '01 8 53 as the E.M.F.
at 20° C. of the E, K, and O series of cells set up with electrolytically prepared paste,
which cells are comparable with the " normal " cell used in our determination.
Allowing for difference of temperature, our value of CxR at 20° C. becomes 1 '01 819,
a difference of 34 parts in 100,000.

As regards the Clark cell, the mean of a number of comparisons made at the
National Physical Laboratory gives the ratio

Clark at 15°C. -r- Cadmium at 17°C. = 1'4066,

* " On a Determination of the Ohm, &c.," by Professor W. E. AYRTON, F.R.S., and Professor
J. V. JONES, F.R.S. ' B.A. Report,' 1897.

t " A New Determination of the Electromotive Force of WESTON and CLARK'S Standard Cells by an
Absolute Electrodynamometer," ' United States Bulletin,' vol. 2, p. 69.

A NEW CURRKNT WEIGHER, KTC.

537

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VOL. covn. — A.

3 z

538 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH :

and using this ratio we get

CxR for Clark cell at 15°G = T4323 ;

GUTHE'S value of CxR for Clark cell at 15°C. = 1-43296.

The difference is in the same direction as that between the cadmium cells, but greater
in proportion.

SECTION 13. — CONCLUSIONS.

From the measurements and observations detailed in the previous pages we may
conclude that the current weigher, constructed on the lines described, is a most
excellent instrument, capable of yielding results of very high precision, and worthy of
acceptance as an international standard instrument for the absolute determination of
the ampere. We therefore hope that other countries will make balances on similar
lines, in order to realise one of the fundamental electrical units in an exact manner.

So far as we are aware, the accuracy attainable by the new balance far exceeds
that secured in any previous absolute determination of any electrical unit, and we
may infer that of all the electrical units the ampere is now the one best known.

Further, we may infer that cadmium cells can be set up so as to be remarkably
constant in E.M.F. The observations made on cell No. 2, set up by one of us
(F. E. S.), extended over a period of 19 months, and during the whole of that period
its measured E.M.F. seldom differed from the mean by more than 1 part in 100,000.

Of the 71 determinations of E.M.F. made —

7 are within 1 in 1,000,000 of the mean,
14 „ 2 „ 1,000,000 „

28 „ 5 „ 1,000,000 „

53 „ 10 „ 1,000,000 „

66 „ 15 „ 1,000,000 „

70 „ 20 „ 1,000,000 „

Only one determination out of the whole 71, and this one of the earliest, differs
from the mean by so much as 1 part in 59,000.

It is of interest to mention that of the 71 determinations just referred to 26 were
made by the same pair of observers (T. M. and F. E. S.), and these show a still closer
agreement, viz., of 26 determinations —

6 are within 1 in 1,000,000 of the mean,

8 „ 2 „ 1,000,000 „
11 „ 5 „ 1,000,000 „
19 „ 10 „ 1,000,000 „
25 „ 15 „ 1,000,000 „

Only the early one previously mentioned differed from the mean by more than

A NEW CURRENT UTIGHER, ETC. 539

1 part in 70,000. The difference between the means of the 26 and the 71 deter-
minations is 1 in 1,000,000.

These results are of considerable importance, as they show very great constancy
both of current weigher and cell. In fact, the cell and balance proved to be much
more constant and reliable than the standard resistance, although the latter was very
carefully made and annealed with a view to ensuring permanency.

The precision of measurement attainable with the new balance exceeds the most
sanguine expectations of its designers. It was intended to give the ampere to 1 in
10,000, and an accuracy of 1 in 20,000 was hoped for, but 1 in 50,000 has been
attained. The instrument itself admits of a far higher accuracy, for a tenth of a
milligramme can be detected with certainty, and this, in a total of 15 grammes, the
balancing mass for 1 ampere, means 1 in 300,000 in the value of the current. This
is a precision considered to be of a very high order, even for relative measurements.
Uncertainty, however, exists as to the value of g, and the axial lengths of the coils,
which prevent the highest accuracy of which the balance is capable, being realised at
present.

Directions in which improvements may be looked for are, therefore :—

(i) A more accurate determination of the acceleration due to gravity, and
(ii) Greater precision in the means for measuring the axial lengths of the coils, or a
lengthening of the coils to reduce the effect of this possible error.

As the uncertainty in g is of most consequence, we trust that an absolute deter-
mination of its value at the National Physical Laboratory will, ere long, be made.

To realise the volt to an accuracy approaching that of the ampere as now known,
it is necessary that an absolute determination of resistance of corresponding precision
be undertaken. At the present time the uncertainty in the absolute value of the
international ohm, in terms of which our values of C x R for the cadmium cells are
expressed, approximates to 4 in 10,000, so it is of considerable importance that a
better determination be made at an early date.

In conclusion, we desire to express our sincere thanks to the British Association for
providing the funds with which to construct the ampere balance, and to Sir ANDREW
NOBLE, F.R.S., for presenting the adjustable stand to support the instrument.

Our most hearty thanks are hereby tendered to Dr. R. T. GLAZEBKOOK, F.R.S.,
Director of the National Physical Laboratory, for supervising the construction of the
electrical portions of the balance, for the keen interest he has taken in the experi-
ments, and also for having placed the very perfect resources of the Laboratory at our
disposal. Indeed, much of the precision attained in the results is due to the facilities
available at the National Physical Laboratory for such work. To Dr. T. E. STANTON
we are indebted for superintending the turning of the marble cylinders used to
support the coils of the balance.

Our best thanks are also due to Mr. J. P. GREGORY for valuable assistance rendered

8 z 2

540 PROFESSOR W. E. AYRTON, MR. T. MATHER AND MR. F. E. SMITH:

in the design, and for the care and skill displayed in making the drawings of the
instrument ; to Messrs. E. FISHER and A. W. HARROLD, late of the Central Technical
College, for checking the initial calculations ; and to Professor A. G. GREENHILL,
F.R.S., for advice concerning the elliptic integrals involved.

APPENDIX A.

Values of coefficients (logs of) and constant terms in series* for complete elliptic
integrals of the first and second kinds (F and E) when k nearly =1. k' = y\ — 1?.

la- 327,4/i 422

*~¥7& \ g'F~i72 3^

I2. 32. 5a7,6/i 4222
+ oi-TT-gi *" ( loS« w - T-o - g-7 ~ g-j

2) * 4: • v A/ I m m O«r± v • 1

+ &c.
This may be written

F (Jb) = log, J + A^^log, J -B2j + A^log. J -B4

+ A6F6(logel-B6)+ &c.
\ » /

Similarly the corresponding series for E (&) may be written

E(i) = l+A/F2(log i-B2')+A/p(logel-B4')

\ K • I \ K /

fcD-TV) + &C.

Values of log AB, log A/, BB, and BB', are given below :—

For F (k).

For E (ft).

n.

Log A,,.

B,,.

Log A,'.

Bn'.

2

I • 3979400

1-0

1-6989700

0-5

4

1-1480625

MO

1-2730013

1-083

6

2-9*97000

1-23

1-0688813

1-20

8

2-8737161

1-269047

2-9317080

1-251190

10

2-7822011

1-291269

2-8279587

1-280159

12

2-7066240

1-306421

2-7444126

1-298846

14

2-6422546

1-317410

2-6744393

1-311916

16

2-5861971

1-325743

2-6142259

1-321577

18

2-5365499

1-332279

2-5613736

1-329011

20

2-4919971

1-337542

2-51427:17

1-334911

* CAYLEY, 'Elliptic Functions,' chiip. III., § 77.

A NEW CUBBBST WEIGHER, ETC. 541

APPENDIX B.

On, the Forces bettveen Coils of Wire of Finite Section*

The formula developed by J. VIRIAMU JONES gives the force when the windings of
the coils can be treated as infinitely tine helical filaments. In the ampere balance,
however, the wires are of finite thickness, and thus small corrections may become
necessary.

If the force parallel to the axis experienced by a helical filament of radius A and of
fixed pitch and number of turns when carrying a current i be F, we have

F=axAd* <"•

where X is the magnetic force at right angles to the axis and B is measured round
the axis. If y be the co-ordinate, parallel to the axis, of one end of the helix, the
force on the helix in a magnetic field symmetrical about the axis is u function of A
and y, and we have

dA* dy* Jo WA* dy* A (/A/

Now, if V be the magnetic potential of the magnetic field, V is symmetrical about
the axis, and hence satisfies LAPLACE'S equation

But X = — </V/</A, and hence, differentiating (3) with respect to A,

_t/X.X

. _

d A' dy> A </A A d A A'
Thus

ePF . d*Y . f'A/X . X\ ,fi 1 dF

T-T-S + -r-r = »l TT + T ) d" = T TT
rfA1 dy* Jo\(/A A/ AJA

-r-r
dy*

This is similar to MAXWELL'S theoremt for mutual induction.

Distribution of Current in the Wire. — In default of any accurate knowledge of the
variations of specific resistance over the cross-section of the wire forming a helical
coil, it is impossible to accurately determine the distribution of current in the wire.
We shall, however, examine the case in which the specific resistance is uniform and
shall call the corresponding distribution of current the " natural " distribution. The
current density at any point may be taken as inversely proportional to the length of

* For the major portion of the following treatment we arc indebted to Mr. Q. F. C. SEARLB, F.R8.
t MAXWELL, 'Electricity and Magnetism,' 3rd ed., vol. ii., § 703.

542 PROFESSOR W. F, AYRTON. MR. T. MATHER AND MR. F. E. SMITH:

one turn of the line of flow through that point. Hence, if the distance of the point
from the axis is A + h, the current density is equal to Ki[(A. + hy +p:t~]~1'2, where K is
a constant, Zirp is the pitch of the helix, and t is the total current through the wire.
To find u, the current per unit area of the section by a plane containing the axis of
the helix, we must multiply the current density by cos a, where a is the slope of the
line of flow. Hence

........ (5).

The constant K is to be determined from the condition that the total current in
the wire is i. When, as in the case of the coils of the ampere balance, p is small
compared with A + h, it will be sufficient to take the first two terms of the expansion
of (5) in powers of p\ and to write

u = Ki{T^-(i£wl'- -" • • • • • <6>-

If h, y be the co-ordinates of the point relative to axes through the centre of the
section parallel and perpendicular to the axis, we can write

h = p cos 4>, y = p sin <j> (7).

Thus, if R is the radius of the wire,

fli rijt rV. riw | i -,2 i

up dp dd> = KM •{ -r r —*- ,r, \ pdp dd).
oJo Jo Jo [A + pcosd> (A + p cos <j>)3 ) *

Now, since p is less than A,

f2* d<f) _ 2:r

and two applications of the reduction formula

P* d<f> 1 A , _£ ^ \ f2' ^

Jo (A+p cos <£)m+1 A\ mdp) Jo (A+pcos <£)m

give

)o(A+pcos<^)3= 2?r 12 (Aa-y)B/3 ~ 2 (A3-/)2)"8]'
On integrating with respect to p, we find

irR.

~A~

A NEW CURRENT WEIGHER, ETC. 543

as far as terms involving R* or R*p*. Hence, to the same order,

If F0 be the force parallel to the axis, experienced by the helical filament defined
by h = 0, y = 0, and if F7 be the force on the helical wire when carrying the same
current, we have

where the force on the helix h, y is expanded by TAYLOR'S theorem. On integration
the first term yields Fu exactly, since the total current is i. For the other terms we
may use (6), and may replace (A + A)"1 by A"1— hA.~t + lt2A.~3 and p*(A. + h)~* by
/>'A~S. When we substitute for h and y from (7) and integrate, we obtain

F_F gKB'f.l/d'F «PF\_ 1 JF 1 *F/R'_j»V. 1 ^F/K'_^\1
A \B\dx* dy'/ 4A</.r0 A'f/o-'Us 8/^P^\16 8/J'

Using (4), and inserting the value of K, we find

F,_F_RV R'-2p'\HF/ R'-2?A RPrPFl
8A\ 4 A* /IcArA 2 A8 / 8A2£j'

or, as far as the terms involving R4 or R*p*,

F'-y R* Ji 3(R'-2j)a)]rfF R4 d'F
"8AI 4A» ]dx. 24Aa(/x08'

This expression includes all the terms up to R4 or R^r arising from the differential
coefficients of not greater than the second order in the Taylor expansion in (9).

In the case of the ampere balance it is unnecessary to go beyond terms involving
R1. To this order we have

F' = F0-^^ .......... (10).

8 A

It is easy to give a physical interpretation to this result. For, if we take a helical
filament of radius A— 2, with its ends in the same planes as the centres of the terminal
sections of the helical wire, the force on it is

The first two terms of this series will be the same as the terms shown in (10) if
2 = R*/8A. Hence, as far as correcting terms involving R3, F' is the force on a helix
of radius A— Ra/8A. Thus, the force experienced by a wire helix of mean radius A
is the same as that experienced by a filamentary helix of radius A — RJ/8A. It is

544 A NEW CURRENT WEIGHER, ETC.

noteworthy that, to this approximation, no correction is to be applied to the axial
length of the coil. The argument applies also to the fixed coils of the balance.

For " natural " distribution of current in the coils of the ampere balance, the
force F calculated on p. 509 is too great. The corrected value of the force is

8ada

where « and A are the mean radii of the suspended and fixed coils. We may
write (11) in the form

aL<M_a.dF , A<M A

" M da '~~~ F ' da q '''' M dA. '' '' F '

where

Inserting the values of R, q, r, A, and a, tabulated on pp. 488, 512, 513, we obtain
a correction to the force of 17 parts in 10,000,000 for the complete system of coils.
The sign of the correction is negative.

If we assume the current density to be uniform, the force in this case is found by
making u = i/TrR3 in (9), and the corrected value of the force is

F

Badet 8A c/A '

The correction is of the same value as before, but of opposite sign. As the distri-
bution of current is uncertain, the value of the force stated on p. 509 lias been used
throughout our work.

[ 545

XIII. The Silver Voltameter.

PART I.— By F. E. SMITH, A.R.C.Sc., and T. MATHER, F.R.S.
PART II.— By F. K. SMITH, A R.C.Sc., and T. M. LOWRY, D.Sc.

(Communicated by R T. GLAZEBROOK, F.R.S.)

Received July 22, — Read November 21, 1907.

(From the National Physical Laboratory.)

[PLATE 9.]

PRINCIPAL CONTENTS.

PART I.

Page

Introduction 546

Description of the voltameters 548

The anodes 649

The electrolyte 549

Form of voltameters 649

Electrical arrangements • 652

Determination of the mass of the deposits 654

Principal table of results 656

Discussion of the result* 562

The Rayleigh form of voltameter 562

The Richards form of voltameter 563

The syphon, pot-syphon-bowl, and syphon-pot-bowl forms 565

The elevated kathode type 566

Efficiency of the porous pote and syphons 566

Deposit on platinum and on silver 567

Influence of pressure 668

The temperature coefficient 669

Liquid inclusions in the deposit 570

Effect of variations in the size of the kathodes 570

Effect of variations in the size of the anodes 671

Variation of potential difference between anode and kathode 572

Variable concentration of electrolyte 672

Variation of current density 673

VOL, CCVII. — A 425. 4 A J12.08

546 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

Page

Effects of electrolysis on the concentration of various portions of the electrolyte 574

The electrochemical equivalent of silver 579

Comparison of results with those of other observers 579

Conclusions 581

PART II.

Preparation of pure silver nitrate 581

Tests of commercial silver nitrate 584

Standard method of preparing silver nitrate solutions for electrolysis 585

Effects produced by repeated electrolysis 586

The question of anodic impurities 587

Examination of the mother-liquors from recovered silver nitrate 589

Striation of the deposit 590

Influence of oxide, carbonate, and chloride 591

Influence of sulphide 592

Influence of nitrite and hyponitrite 593

Influence of acids 594

Effect of heating silver nitrate 596

Electrolysis of a solution of silver acetate 597

Electrolysis of a solution of silver chlorate 597

Electrolysis of a solution of silver perchlorate 597

Summary 598

INTRODUCTION.

IT has been known for several years that the measurement of electric quantity by
the electro-deposition of silver is liable to inaccuracies which appear to be dependent
on the condition of the anode, kathode, and electrolyte of the voltameter (or coulo-
meter) employed. In 1884 Lord RAYLEIGH* and Mrs. SIDGWICK found that a small
quantity of silver acetate added to a solution of silver nitrate or of silver chlorate
apparently increased the mass of silver deposited per ampere-second, and in 1895
RODGER and WATSONt showed that the silver voltameter was liable to give results
varying by as much as 1 part in 1000 when the same solution of silver nitrate was
repeatedly used. The latter effect was thought to be due to the formation of a
complex silver salt, and in 1899 RICHARDS, COLLINS, and HEIMROD| practically
confirmed this view. A new form of silver voltameter was suggested by the latter
observers, and this has been frequently used for observations of precision. In 1898
K AHLE § made a very large number of measurements, using platinum and silver bowls
as kathodes and silver nitrate solutions, treated in various manners and from many
sources, as electrolytes. He found that the deposit of silver per colomb was greater
on a silver surface than on one of platinum ; that it increased with the continued use

* RAYLEIGH and SIDGWICK, 'Phil. Trans.,' 175, p. 411, 1884.

t RODGER and WATSON, 'Phil. Trans.,' A, 186, p. 631, 1895.

J RICHARDS, COLLINS, and HEIMROD, 'Proc. Am. Ac.,' 35, p. 123, 1899.

§ KAHLE, ' Zeitschr. Inst.,' 18, pp. 229-267, 1898.

ON THE SILVER VOLTAMETER. 547

of a solution, and that the nature of the deposit also varied with the solution
employed. In 1892 SCHUSTER and CROSSLEY* discovered that the mass of silver
deposited was related to the pressure and also to the size of the silver anode ; the
pressure effect was verified by KAHLE,! RICHARDS,! and MYERS,§ and the latter
observer found an increase when the liquid was saturated with nitrogen, but a
decrease when the dissolved gas was carbon dioxide. MERRILL || repeated the pressure
experiments and found no effect due to change of pressure alone. Lord UAYLEIOH
and Mrs. SIDOWICK observed an increase of deposit with increase of temperature ;
RICHAHDS, COLLINS, and HEIMROD obtained greater deposits at 60° C. and at 0° C.
than at 20° C. LfiDucIT found a decrease with increasing temperature, aud MKI:I:I i.i.
suggests that the mass is independent of the temperature.

In more recent years GUTHK** and VAN DiJKtt have made a special study of
various forms of voltameters. The form suggested by RICHARDS was found by him
to give a smaller deposit of silver than the form originally devised by Lord
RAYLEIOH. The difference between the two forms found by RICHARDS in 1899 was
80 parts in 100,000; in 1902 he found 44 parts in 100,000; WATSON,}} in 1901,
obtained a difference of 26 ; GUTHE, in 1904, found 48, and in the same year
VAN DIJK ol>served a difference of 23. VAN DIJK also compared the syphon and
Rayleigh types and found a mean difference of 8 parts in 100,000, the latter form
giving the heavier deposit ; if a very doubtful observation is excluded, the mean
difference is 18 parts in 100,000. In addition, VAN DIJK observed a difference due to
the size of the platinum bowls, the smaller one invariably containing the lighter
deposit for the same form of voltameter. There are many other interesting differences
which need not now be enumerated ; sufficient has been written to show that the
silver voltameter could not be regarded as an instrument of high precision. The
international ampere is, however, defined in terms of the deposit of silver, and the
Conference on Electric Units at Charlottenburg in October, 1905, reaffirmed this
definition, but expressed the opinion that the information before it was not sufficient
to enable it to lay down exact directions in respect to the silver voltameter to be
employed. Hence the necessity for an enquiry to ascertain the possibility of specifying
a voltameter which is easily reproducable and in which an ampere-second always
deposits the same mass of silver.

* SCHUSTER and CROSSLEY, ' Roy. Soc. Proc.,' 50, p. 344, 1892.
t K.UII.K, 'Brit. Assoc. Report,' Section A, 1892.
I RICHARDS and HEIMROD, 'Proc. Am. Ac.,' 37, p. 415, 1902.
§ MYERS, ' WIED. Ann.,' 55, p. 288, 1895.
|| MERRIIX, 'Phys. Rev.,' 10, p. 167, 1900.
f LEDUC, ' Journ. de Phys.,' 1, p. 561, 1902.

** GITIIK, • 1'hys. Rev.,' 19, p. 138, 1904. 'Bull. Bureau of Stands.,' vol. 1, No. 1, p. 28, 1906.
tt VAN DIJK and KUNST, 'Ann. der Phys.,' 14, p. 569, 1904. VAN DIJK, 'Ann. der Phys.,' 19,
p. 249, 1906.

tt WATSON, ' Phil. Trans.,' 1898, p. 445, 1902.

4 A 2

548

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

For convenience this communication is divided into two parts. In some of the
very early experiments it was ftmnd that the mass of the deposit was dependent on
the mode of preparation of the silver nitrate. A considerable quantity of the
crystallised salt was necessary for the observations, and for economy and experience
the salt was recovered from the used solutions by recrystallising. The first products
gave very remarkable results, both as regards the deposited mass and its appearance,
but on further purification the normal salt giving normal deposits was obtained. The
abnormal results were found to be due to impurities hitherto unsuspected, and which
are hot usually contained in the salt purchased as pure in the ordinary way ; they
appear to arise in the manipulation of the solution, the fusion of the salt, and its long
exposure to the air. These impure solutions are dealt with in the second part of this
communication. In the measurements discussed in the first part the pure salt only
was used.

PAKT I.

A Comparison of Various Forms of Silver Voltameters, by F. E. SMITH, A.R.C.Sc. ;

and a Determination of the Electrochemical Equivalent of Silver,

by F. E. SMITH, A.R.C.Sc., and T. MATHER, F.R.S.

Description of the Voltameters.

Eleven platinum vessels and two silver ones have been used for the kathodes of the
voltameters. The dimensions, approximate masses, capacities, &c., of these vessels
are given in the following table. We are indebted to Messrs. Johnson, Matthey
and Co. for the loan of the vessels A, B, C, D, L, and M.

Letter
by which the
vessel is
indicated in
Table I.

Shape.

Approxi-
mate mass
in grammes.

Diameter at
mouth, in
centimetres.

Depth,
in
centimetres.

•

Maximum

capacity,
in cubic
centimetres.

Convenient
volume of
electrolyte,
in cubic
centimetres.

A

Bowl

80

10

8-2

570

300-450

B

80

10

8-2

570

300-450

C

83

10

8-2

570

300-450

D

87

10

8-2

570

300-450

E

78

10

3-8

250

150-200

F

30

10

4-5

280

150-200

O

30

10

4-5

280

150-200

(Silver) HS

85

10

6-5

450

300-400

(Silver) KS

90

10

6-5

450

300-400

L

65

6-3

6-0

170

100-130

M

Ring

65

7-7

5-7

—

N

Bowl

39

6-0

4-1

75

50

0

Crucible

39

4-1

5-0

50

40

ON THE SILVER VOLTAMETER. 549

L, M, and N are platinum vessels with lateral projecting ears to support them
during electrolysis. L is nearly cylindrical in shape, M is a cylindrical ring only, and
N is hemispherical. The vessels N and 0 were very kindly lent to the National
Physical Laboratory by Professor G. VAN DIJK ; N is the small bowl referred to as B
in Professor VAN DIJK'H papers,* and O is one of the crucibles indicated by L and II.
For the vessel E we are indebted to Professor AYKTON.

The Anodes.

An anode usually consisted of a pure silver plate, 5 'Ox 5 '0x0 '4 centims., held by
a silver rod riveted through its centre. The silver was obtained from Messrs.
Johnson, Matthey and Co. Before employing it for a determination of the electro-
chemical equivalent of silver, about 10 grammes of silver were deposited electrically
on the plate, and on these occasions a platinum bowl with a deposit of silver on its
inner surface was generally used as the anode. In this way the silver was always
removed from the platinum bowls, but not from those of silver. When new, the
surface of a platinum vessel is free from scratches, and silver deposited on it adheres
much better than to a scratched surface ; it is important, therefore, to avoid the use
of a spatula. The electrical method removes the deposit, cleans the platinum, and
at the same time prepares the anode. Shortly before using, the latter was washed
with distilled water and dried in an electric oven. The platinum bowls were rinsed
with distilled water and strong nitric acid, and if much of the dark silver salt
Ag7NOu adhered to the platinum the washing with nitric acid was repeated.
Distilled water was finally used and the bowls dried in an electric oven at 1 60° C.

For the Richards form of voltameter silver rods of two sizes were used ; some of
the rods were 1 centim. in diameter and the others 2'5 centims. These rods were also
coated with electrically deposited silver.

The Electrolyte.

For the comparison of various types of voltameters the electrolyte was usually a
15 per cent, solution of pure silver nitrate in water. The silver nitrate was sometimes
purchased and sometimes recovered from used solutions. A description of the
preparation of the pure salt is given in Part II., p. 585, of this communication.

Form of Voltameters.
The following types were used :—

(1) The Rayleigh Form (fig. l). — In this the kathode was usually a platinum bowl
and the anode a silver plate or rod coated with electrically deposited silver. The
anode was inserted in a cup made of filter paper, the folds in the paper being secured

* VAN DIJK and KUXST, 'Ann. der Phys.,' U, p. 569, 1904. VAN DIJK, 'Ann. der Phys.,' 19,
p. 249, 1906.

550

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

with a little sealing-wax. The silver plate was immersed in the solution just below
the surface, and the sealing wax which secured the folds of the filter paper was at
least 2 centims. above the surface of the electrolyte. The filter cup was supported by
three platinum wires from an ebonite ring.

Fig. 1.

Fig. 2.

(2) The Richards Form (fig. 2). — A platinum bowl was in general used for the
kathode, and the anode was of the same form as that of the Rayleigh voltameter. It
was, however, surrounded by a fine-grained porous pot instead of a filter paper, and
the electrolyte inside the pot was maintained at a lower level than the electrolyte in
contact with the kathode. Usually no appreciable change in the difference of level
resulted during an experiment, but a small syphon was frequently employed to ensure
an approximately constant difference. Two of the porous pots were made by the Akron
Insulator and Marble Company, of the United States of America, and were obtained
for us by Dr. GUTHE ; several were made by PUKAL, of Berlin ; and others were from
an unknown source.

(3) The Syphon Form (fig. 3). — The kathode was a platinum bowl and the anode
a silver plate or silver bowl coated with electrically deposited silver. When a silver

Fig. 3.

Fig. 4.

plate was used, it was contained in a glass dish, the electrolyte in the latter being
connected with that in the platinum bowl by a glass syphon. Two sizes of syphon
were used ; one was 30 sq. centims. in its narrowest part and 30 centims. in axial
length, the other was 8 sq. centims. section and 20 centims. in axial length.

(4) The Pot-Syphon-Bowl Form (P.S.B.} (fig. 4). — In this the liquid in the syphon
was separated from that in the anode vessel by a porous pot. Any advantages of the
Richards form were thus combined with those of the syphon.

ON THE SILVER VOLTAMK TKl:.

551

(5) The Syphon- Pot- Bowl Form (S.P.B.) (tig. 5). — In this the liquids in the syphon
aud kathode vessel were separated by a porous pot. Any disadvantages of the
Richards form were thus introduced without eliminating any advantage of the
syphon.

Fig. 5.

Fig. 6.

(6) The Elevated Kathode Form (fig. 6). — A silver bowl with electrically deposited
silver formed the anode, and a platinum ring, or platinum bowl of smaller diameter
than that of the silver bowl, formed the kathode. If a heavy anode liquid was
formed, it would not come into such intimate contact with the kathode as in the
Rayleigh form.

(7) Several Modifications of the above. — (a) The silver anode in the Rayleigh type
was replaced by a platinum anode ; (b) for the filter paper of the Rayleigh form a
porous pot drilled with tine holes was substituted, and purified asbestos was placed in
the pot to prevent any anode slime reaching the kathode ; (c) in addition to the filter
paper a china filter cup with very fine holes surrounded the anode of the Rayleigh
form. The internal resistance was thus increased 100 times, and the potential
difference between anode and kathode was correspondingly increased.

In all of the forms the platinum bowls used as kathodes were supported on brass
rings mounted on ebonite, and the silver rods supporting the anodes were clamped to
a metal arm projecting from a rod similarly mounted. From 300 to 400 cub. centims.
of solution were used in the large bowls and about 30 to 40 cub. centims. in the
small crucible loaned to us by Professor VAN DIJK. The solution was introduced by
means of a pipette and was similarly removed after the required amount of silver was
deposited. The liquid was carefully examined for loose silver, and if any was found,
the solution and the water used for washing the deposit were filtered through a hard
filter paper, the particles of the silver washed to the lowest part of the paper, and
the latter dried in an electric oven. By the aid of a pointed glass rod the loose
silver was transferred to the bowl. The main portion of the deposit was washed by
rinsing three or four times with distilled water, after which the bowl was filled with
water and left overnight. This last wash-water rarely showed more than the faintest
bluish colour on addition of neutral NaCl solution. Two more rinsings with water
followed, and then the bowl was dried by heating in an electric oven at 160° C.

552

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

Electrical Arrangements.

With the exception of that portion of the circuit which included the silver
voltameters, the apparatus and its arrangement was the same as that used in the
determination of current in absolute measure by the British Association (Ayrton-
Jones) Ampere Balance.* The current which it was necessary to pass through a
standard resistance at a known temperature in order that the potential difference on its
terminals just balanced the E.M.F. of a Weston cadmium cell at a constant tempera-
ture, was determined by the balance, and for other slightly different temperatures
of the coil and cell the change of current was calculated from their temperature
coefficients. Neither cell nor coil was assumed to remain constant except over very
short periods of time, and as the change in E.M.F. of the cell was determined to be
not greater than 1 part in 100,000 in eighteen months and the secular change of the
resistance coil was easily determined by comparison with the National Physical
Laboratory Standards, no appreciable error was thus introduced. The probable error
of the measure of the current in absolute units is shown in the communication dealing
with the ampere balance to be about 2 parts in 100,000, and the mass of silver
deposited per ampere-second in the silver voltameter is subject to the same error.

Fig. 7.

The arrangement of the apparatus is represented by fig. 7. The current was
furnished by a battery B of 55 accumulators of 30 ampere-hours capacity, and in
series with it were placed a three-dial adjustable resistance R! of 111 ohms,
a double-groove mercury trough M for fine adjustment of the current, a standard
resistance R of manganin strip built to carry a current of 10 amperes, an ammeter A,
and the voltameters V. The latter were put in and out of the circuit by the
switch K. This was specially designed to close the circuit of a chronograph at the
same time as that of the voltameters and to close it again when the latter circuit

* AYRTQN, MATUER, and SMITH, ' Phil. Trans.,' A 207, p. 518, 1908.

ON THE SILVER VOLTAMETER. 553

was opened. The precision with which this was secured was tested by placing a
second chronograph and battery in place of the silver voltameters and noting the
difference in the intervals of time recorded by the two instruments. The mean of
20 readings indicated a difference of a little less than one-hundredth of a second,
which is equivalent to an error of 1 part in 600,000 in the observations made with
the voltameters. The time was measured by the standard clock presented to the
National Physical Laboratory by Lady GALTON. The rate of the clock was deter-
mined by means of signals from Kew and Greenwich. The battery was earthed at
such a point, E, that the mean difference of potential between the voltameters and
the earth was very nearly zero, but the insulation of all the apparatus from earth was
also very carefully attended to. The switches C and CX were on one board and
could not be separately operated ; C reversed the current through the standard
resistance, and C7 reversed the connections of the standard cell to the potential points
of the resistance. C" is a switch for placing either of two cells S, S' in the potentio-
meter circuit ; S' was employed for the adjustment of the current before including the
voltameters in the circuit, and S continually afterwards. No secondary potentiometer
circuit was used. The resistance coils of RI were of manganin and were immersed in
a large bath of paraffin oil. The double mercury trough M was bridged by a copper
sliding piece which shunted a portion of the resistance of the trough and allowed ot
a fine adjustment. A change in current of 1 part in 1,000,000 was easily detected,
and sometimes a current constant to this amount could be maintained for an hour or
more. G was a galvanometer of the Broca type of 1000 ohms resistance.

The circuit was at first closed so as to exclude the voltameters, and remained
closed for 1 hour or more before any adjustment for constancy of current was made.
On many occasions a determination of current in absolute measure preceded the
deposition of silver. This usually occupied 20 minutes ; immediately afterwards R,
was diminished by an amount comparable with that of the voltameters and the latter
switched into the circuit. With the Rayleigh form of voltameter a current steady to
1 part in 100,000 was secured within 20 seconds after closing the circuit ; a slightly
longer time was necessary for the Richards type and longer still for the largest of
the syphons.

Owing to the difficulty of maintaining a steady current through the syphon and other
modified forms of voltameters, some of the observations are relative only. In these
latter cases the standard is the Rayleigh form, but the constancy of this had been
well established before any relative observations were made. In order to distinguish
between the relative and the absolute values, we have placed an asterisk against all
absolute determinations.

When Lord RAYLEIGH* determined the electrochemical equivalent of silver, the
current that passed through the voltameters also passed through the standard current
lalance, and was thus directly determined in absolute measure. We also might have
* RAYLEIOU and SIDGWICK, 'Phil. Trans.,' 175, p. 411, 1884.

VOL. ccvn. — A. 4 B

554

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

adopted this method, but, owing to the ampere balance being less steady when a
current passes through it for more than half an hour, the measurement of the current
through the voltameters would have been subject to a comparatively large error.
By frequently standardising a combination of resistance coil and cadmium cell (as
already described) and employing these for the measurement of ciirrent through the
voltameters, the probable error was appreciably reduced.

Mass of Depo * //*.

The large bowls were weighed on a balance by L. OERTLING, sensitive to one-fiftieth
of a milligramme ; the small bowls were weighed on a more sensitive balance by
OERTLING, and very kindly lent by Mr. OERTLING for this work, or on a balance by
RUPRECHT, which was loaned to the National Physical Laboratory by Dr. SCOTT, of
the Davy-Faraday Laboratory. All weighings were made by the method of GAUSS,
a similar vessel being used as a tare. The mass of silver usually deposited was about
7 grammes in the large bowls and a single weight of 7 grammes was used to counter-
poise ; the difference was obtained by means of a rider. The weight employed was
standardised by comparison with a 100-gramme weight from Sevres. For difference
determinations the four bowls A, B, C, D were largely used.

To reduce the error of weighing we have found it convenient to compare the
masses of the bowls when empty and again with the deposits of silver. The following
example is one with the bowls containing silver, and gives an idea of the error
introduced : —

Observation.

Bowls.

Difference in mass
in grammes.

Calculated from
observations

Difference in mass.
Mean value in grammes.

1

AandB

+ 0-49678
682
676

2 and 4
3 „ 5

+ 0-49679

2

A „ C

-3-94383

387
389

1 „ 4
3 „ 6

-3-94386

3

A „ D

-6-58421
419
415

1 „ 5
2 „ 6

-6-58418

4

B „ C

-4-44065
061
065

1 „ 2
5 „ 6

-4-44064

5

B „ D

-7-08097
099
097

1 „ 3

4 „ 6

-7-08098

6

C „ D

-2-64032
038
032

2 „ 3
4 „ 5

-2-64034

ON THE SILVER VOLTAMETER. 555

It is of interest to state that the diminution in mass of these four bowls from
June, 1906, to June, 1907, is 0'8 milligram only. The electrical method of removing
the deposit from the bowl does not take the platinum into solution, and the mass of
any one bowl in a series of five or six experiments remains constant to O'l milli-
gramme. We observed this constancy and found it a most useful check on our
weighings of the empty vessels.

As the results of our observations differ very materially from those of nearly all
other workers on the same subject, we include particulars of the kathode, anode,
solution, current and deposit in Table I. These results are in chronological order and
include the measurements discussed in Part II. of this paper.

When two or more voltameters were placed in series and the same current primed
through each, the results of the observations are indicated by the same number, but
by different letters. For those observations in which the current was maintained
steady throughout the time of the deposit, the value of the current in amperes
(10~! C.G.S. unit) is stated to 1 part in 1,000,000. This is for the accurate com-
parison of results ; the error of this measure is about 2 parts in 100,000. The time
is recorded to one-hundredth of a second and is probably correct to O'l second. When
the observations were made merely for the comparison of different forms of volta-
meters and the current was not maintained steady, the approximate value of the
current only is given. Except in four special cases (43a, b, c, d) the electrochemical
rijiiivalent is calculated to 1 part in 100,000.

Under the heading of solution, letters are given which indicate the source of the
silver nitrate crystals ; H, M, G, and W are samples of silver nitrate purchased
from four different manufacturing chemists, and when more than one sample was
purchased from the same firm a number accompanies the letter. LR indicates that
the salt was recrystallised by LOWRY, and SR that the same process was conducted
by SMITH. WS is a solution from Professor WATSON, and the VD solutions were
prepared from salt recrystallised by. Professor VAN DIJK. The degree of electrolysis
of the solution, i.e., the ratio of the silver previously deposited by the passage
of an electric current through the solution to that present in the solution is stated
in column 8, and some brief notes on the appearance of the deposits appear in the
last column. The pressure to which the electrolyte was subjected and its temperature
are also stated, but only when these differed from the pressure and temperature of the
air of the room in which the observations were made.

In column 4, R denotes the Rayleigh form of voltameter, P the Richards (porous
pot) form, S the syphon form, P.S.B. and S.P.B. the arrangements we have called the
pot-syphon-bowl, and syphon- pot-bowl forms, respectively, and EL.K. the elevated
kathode form.

4 B 2

556

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

Remarks on deposit,
ssure, temperature, A

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ON THE SILVER VOLTAMETER.

557

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MR. F. E. SMITH, MR. T. MATHKK, AND DK. T. M. LOWKY

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560

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VOL. OCVII. — A.

4 0

562

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

Discussion of the Results.

Observations 1 to 13 were made with solutions which are now known to have been
impure. Apart from the electrolytes, the observations are also very unsatisfactory,
for in these early experiments the silver deposits were partly removed with the aid of
a platinum spatula, and the remainder with warm nitric acid ; the bowls were much
scratched in consequence, and many of the deposits were very loose. In addition, we
were not so expert in the manipulation of the voltameters as in the later experiments,
although the errors introduced on this account are regarded as very much smaller
than the differences in the electrochemical equivalent which we found from day to day.
As the electrolytes were not pure we have deferred the discussion of the first 13
sets of observations to Part II. (pp. 582, 595).

The Rayleigh Form of Voltameter.

Table II. summarises the results obtained with the liayleigh form. In all, there
are 52 independent observations ; the mean value of the electrochemical equivalent is

TABLE II. — Rayleigh Form. Absolute Determinations.

No.

Milligrammes
per

Difference
from

No.

Milligrammes
per

Difference
from

No.

Milligrammes
per

Difference
from

coulomb.

mean.

coulomb.

mean.

coulomb.

mean.

Ha

1-11827

0

22c

1-11825

-2

43a

1-118269

0

146

25

— 2

23d

32

+ 5

436

264

— 1

15a

26

- 1

24a

27

0

43c

258

- 2

156

22

- 5

246

25

-2

43d

275

0

16a

25

_ 2

25d

28

+ 1

44a

27

0

16ft

22

- 5

27a

28

+ 1

446

26

- 1

17a

22

- 5

27c

28

+ 1

44c

16

-11

176

33

+ 6

2ld

30

+ 3

466

32

+ 5

17c

27

0

28a

33

+ 6

46c

30

+ 3

\1d

25

- 2

286

28

+ 1

47a

25

- 2

18a

38

+ 11

28c

26

-1

51a

23

- 4

186

40

+ 13

29a

27

0

796

25

- 2

18c

26

- 1

296

29

+ 2

84«

I':.

- 2

19

29

+ 2

30«

27

0

846

28

+ 1

20

25

- 2

305

27

0

86a

26

- 1

21a

34

+ 7

30^

23

— 4

866

26

- 1

216

29

+ 2

31a

28

+ 1

21e

30

+ 3

346

28

+ 1

Mean = 1-118273 milligrammes per coulomb.

1 '118273 milligrammes per coulomb, and the mean observational error is 2'4 parts in
100,000. The greatest differences from the mean occur with 18«, 186, and 44r, and it
is possible that the solution used on the occasions 18a and 18ft was impure owing to

OS THE SILVER VOLTAMETER. .r,;;

its prolonged contact with the atmosphere. If we exclude these results the mean is
ril826g. The mean difference of 2'4 parts in 100,000 must not be taken as the
figure indicating the reproducibility of the Rayleigh form, for the errors of time, of
current determination, of weighing, and of manipulation, are factors in this, as well
as the ]X)ssible changes which ensue due to slight variations in anode, kathode and
electrolyte, and which jointly constitute the error associated with the voltameter. If
we exclude all of the errors except those due to the voltameter and faulty manipula-
tion, the mean difference is very small ; this is well illustrated in observations
43a, b, c, </, the results of which are given to 1 part in 1,000,000. The mean of
these four results is ril826s, and the mean difference is only 7 parts in 1,000,000,
which is probably much lower than the usual error of manipulation. We feel
justified, therefore, in regarding the Rayleigh form of voltameter, as employed by us,
to be reproducible within 1 part in 100,000.

The Richards Form of Voltameter.

The results obtained with the Richards form were at first more variable and always
lower than when the Rayleigh form and the syphon form were employed. For some time
we were at a loss to understand why the Richards value should be lower than that ot
the syphon, for both forms do, to a considerable extent, exclude the anode liquid from
the kathode vessel, and the changes in concentration of the kathode liquids are also
comparable. We eventually found the discrepancy to be due to the porous pots, of
which we had three kinds : — (1) From the Akron Insulator and Marble Company of
the United States ot America ; (2) from PUKAL of Berlin ; (3) from an unknown
source. Dr. GUTHE kindly obtained the pots (1) for us, and they are similar to
those used by him in his research on the Silver Voltameter at the National Bureau of
Standards.* The second type of pot is larger, but presumably of the same kind of
ware as the pots used by RICHARDS. The third class of pot is from an unknown
source ; they were made from large porous pots obtained through the agency ot
Messrs. W. & J. GEORGE, Ltd.

We cleaned the pots with aqua regia, potassium cyanide, nitric acid, and hot
distilled water before using in the voltameter. After a few runs they became
stained, and further cleaning with potassium cyanide, nitric acid, and water was
necessary. In our earlier experiments the pots were soaked in several lots of distilled
water for 24 hours before using, and in neutral silver nitrate solution for 3 or more
hours before the erection of the voltameter. The water in which the pots were finally
soaked was invariably free from acid sufficient to redden very sensitive blue litmus
paper, but we were forced to conclude that the pots were not acid-free, for on
electrolysis of silver nitrate with a pot interposed between anode and kathode the
solution in contact with the kathode became sufficiently acid to affect litmus. An

* GUTHE, • Phys. Rev.,' 19, p. 138, 1904 ; ' Bull. Bureau of Stands.,' vol. 1, No. 1, pp. 28 and 349, 1904.

4 C 2

564

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

acid solution usually gives a lower deposit of silver than a neutral one (p. 595), and in
consequence the results with the porous pot form were at first more variable and
lower than with the Rayleigh form. In addition, some cyanide may have been
present in solution. More consistent results were obtained on prolonged soaking, but
we found the most satisfactory treatment was to place the pot in an electric furnace
for a few hours. This procedure was adopted in our later observations, and the mean
of the values resulting from these is given separately (Tables III. and IV.). The
final result, 1 '11828, is practically the same as that obtained for the Rayleigh form,
and we conclude that the porous pot in our form of voltameter is of no advantage.

TABLE III. — Richards Form. Absolute Determinations.

No.

Milligrammes
per coulomb.

Difference from
mean.

No.

Milligrammes
per coulomb.

Difference from
mean.

22a
34e
34d
44d
46a

l-11797t
809t
05f

oot

16t

- 8
+ 4
0
- 5
+ 11

*

46rf
79a
79c
79rf

1-11831*
26*
30*
25*

+ 3
-2
+ 2
-3

Mean of * observations — 1-11828 milligrammes per coulomb.
Mean of t 1- 11805

* Porous pots baked in electric furnace after soaking in water.

t Porous pots soaked in acid and afterwards in water for several days.

TABLE IV. — Comparison of the Richards Form with the Rayleigh Form. The latter
is taken as the Standard (1 Coulomb deposits I'll 8 27 Milligrammes of Silver).

No.

Milligrammes
per coulomb.

Difference from
mean.

No.

Milligrammes
per coulomb.

Difference from
mean.

32a

M1813t

-7

49e

1-11827*

-1

396

20t

0

576

27*

-1

41a

20t

0

64c

30*

+ 2

41rf

20t

0

696

28*

0

42a

22f

+ 2

69c

28*

0

42c

22t

+ 2

456

21t

-f-1

45c

20t

0

\

Mean of * observations = 1 -11828 milligrammes per coulomb.
Mean of t 1-11820 „ „

* Porous pots baked in electric furnace after soaking in water.

t Porous pots soaked in acid and afterwards in water for several days.

ON THK SILVER VOLTAMETER.

That the porous pots were sufficiently close-grained to keep the anode and kathode
li«|iiiils apart is shown on p. 566. The results obtained with pots soaked in nitric
acid, and afterwards in- several lots of distilled water for about 2 to 4 days, an
given in 22a, 34r, 34d, 44rf, 46a, 32a, 396, 4 la, 4 id, 42a, 42c, 456, and 45c.
These values are in fairly good agreement with one another, but all are lower than
the figure obtained for the Rayleigh form.

The Syphon, Pot-Syphon-Bowl (P.S.B.), and Syphon-Pot-Bowl (S.P.B.) -Forms

(Table V.).

The results for the syphon form and for the P.S.B. and S.P.B. types are mainly
comparative, the Rayleigh form being taken as the standard. The P.S.B. and S.P.B.
types give the effect of the porous pot and clearly indicate that the low results of the
early experiments with the Richards form were due to contamination of the solution

TABLE V. — Comparison of the Syphon, Pot-Syphon-Bowl, and Syphon- Pot-Bowl
Voltameters with the Rayleigh Form. The latter is taken as the Standard
(1 Coulomb deposits Til 827 Milligrammes of Silver).

Syphon.

P.S.B.

S.P.B.

No.

Milligrammes
per
coulomb.

Difference
from
mean.

No.

Milligrammes
per
coulomb.

Difference
from
mean.

No.

Milligrammes
per
coulomb.

32c

1-11825

-2

336

1-11825

-3

35at

1-11819

:;'.'.•

28

+ 1

356

31

+ 3

36at

01

42</

29

+ 2

366

31

+ 3

37«*

25

4M

29

+ 2

376

27

-1

406*

28

51et

25

-2

40e

24

-4

55r

31

+ 4

45</

30

+ 2

57e

25

-2

Ml

28

+ 1

27

0

Mean = 1-11827

Mean =1-11828

Mean of * = 1-11826S
Mean of t = 1-11810

* Porous pots baked in electric furnace after soaking in water.

t Porous pote soaked in acid and afterwards in water for several days.

; Al>solute determination.

by the liquid included in the pot. The result of observation 51c is not comparative
(see Table I.), the very large syphon being used on this occasion. The mean value of
the electrochemical equivalent is practically the same for the three forms, and agrees
also with the values obtained with the Rayleigh and Richards types of voltameter.

5fi6

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

Elevated Kathode Type.

The results are given below (Table VI.) and call for no comment. No difference
from the usual type was anticipated, for the Rayleigh and syphon forms had been
found to agree before the first of the observations in Table VI. had been made.

TABLE VI. — Comparison of the Elevated Kathode Voltameter with the Rayleigh

Voltameter (I'll 827).

No.

516*
51rf*
82c

Milligrammes per
coulomb.

1-11829
26
24

Difference from
mean.

+ 3

0

Mean = 1-11826
* Absolute determinations.

Efficiency of Porous Pots and of Syphons.

Nearly all previous researches have led to the conclusion that a source of trouble
exists at the anode, and the porous pot voltameter was designed by RICHARDS to
prevent the anode liquid having access to the kathode. NOVAK* was the first to
suggest the possible existence of a complex silver salt in the electrolyte ; RODGER
and WATSON! independently made the same suggestion ; RICHARDS'^ experiments
strengthened the view, and the work of KAHLE,§ GUTHE,|| VAN DiJK.1I and others
supports the theory of anode complications. When the very excellent work of these
observers is reviewed the results of our observations are astonishing, and we deem it
necessary to give evidence of the reliability of our porous pots and syphons.

When the porous pots of classes (1) and (2) were filled with water and allowed to
stand for 12 hours, about 40 per cent, of the liquid appeared to pass through their
walls. When pots of class (3) were similarly treated, less than 5 per cent, of the
liquid passed through the pots. In one of the pots of class (3) 50 cub. centims. of the
electrolyte used in observation 30c, which gave a deposit of 1 '12055 milligrammes
per coulomb, were used for the anode liquid in observation 49c, the kathode liquid
being normal AgNOs solution. The result is 1*11827, showing that very little of the

* NOVAK, 'Proc. Roy. Bohemian Ac. ScL Prague,' 1, pp. 387-432, 1892.

t RODGER and WATSON, ' Phil. Trans., ' A, 186, p. 631, 1895.

t RICHARDS, COLLINS and HEIMROD, 'Proc. Am. Ac.,' 35, p. 123, 1899.

§ KAHLE, 'Zeitschr. Inst.,' 18, pp. 229-267, 1898.

|| GUTHE. 'Phys. Rev.,' 19, p. 138, 1904; 'Bull. Bureau of Stands.,' vol. 1, pp. 28 and 349, 1904.

U VAN DIJK and KUNST, ' Ann. der Phys.,' 14, p. 569, 1904 ; VAN DIJK, 'Ann. der Phys.,' 19, p. 249, 1906.

OH Till: SILVER VOLTAMETER.

56,

abnormal liquid could have diffused through the walls of the pot. An analysis of
the strength of the anode and kathode solutions before and after electrolysis was
made in other experiments and led to the conclusion that the pots were efficient.
The syphon was tested by placing the electrolyte used in observation 30c in the anode
lx>wl and a normal solution in the syphon limb and kathode bowl ; the result,
I'll 829 (49d), shows that no appreciable quantity of the anode liquid could have
diffused into the kathode vessel.

Deposit on Platinum and on Silver.

Table VII. gives the results when silver bowls were employed as kathodes. Our
successful employment of these bowls is in a large measure due to the use of an
electric oven for drying purposes.

TABLE VII. — Comparison of Results with Platinum and Silver Kathodes. The Mean
Value of the Results with Platinum Bowls is I'll 827 Milligrammes of Silver per
Coulomb.

Results with Silver Bowls as Kathodes.

No.

Milligrammes
per

Difference
from

No.

Milligrammes
per

Difference
from

No.

Milligrammes
per

Difference
from

coulomb.

mean.

coulomb.

mean.

coulomb.

mean.

146

1-11825

- 3

22e

1-11825

-3

:;«•,

1-11827

-1

15a

26

- 2

23d

32

+ 4

306

27

-1

16a

25

- 3

•j.v

28

0

31a

28

0

17a

22

- 6

26a

30

+ 2

32c

25

-3

176

33

+ 5

266

27

-1

346

28

0

186

40

+ 12

27a

28

0

64rf

28

• o

tie

30

+ 2

Ml

33

+ 5

•Mean - 1-11828

It is apparent that the deposit on a clean platinum surface is the same as that on a
silver surface. This result is in agreement with VAN DIJK'S* oljservations.

GORE, KAHLE,! and RICHARDS and HEIMROD^ found a somewhat larger deposit
when the kathode of the Rayleigh form was of platinum with silver deposited on it
than when the kathode at the commencement of the observation was platinum only.
RICHARDS found that the deposit on a silver kathode was 1 part in 10,000 heavier

* VAX DIJK, 'Ann. der Phys.,' 19, p. 282. 1906.

t KAHLK, 'Zeitechr. Inst,' 18, pp. 229-267, 1898.

} RICHARDS and HEIMROD, 'Proc. Am. Ac.,' 37, p. 418, 1902.

568

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWIIY

than that on one of platinum when the Rayleigh form of voltameter was used, but
that the masses were equal for the porous pot form. GUTHE* confirmed the latter
result.

Influence of Pressure.

In testing for a pressure effect, one of the voltameters was placed under a glass
bell jar in which the gaseous pressure could be varied from 2 '5 centims. of mercury to
atmospheric pressure. The voltameter was supported on a cast-iron plate through
which the leads passed, the latter being well insulated from the plate by ebonite and
rubber, and tests made before and after each experiment proved the efficiency of the
insulation. In two cases the solution under diminished pressure was made with silver
nitrate and distilled water which was boiled a few minutes before setting up the
voltameter ; in the remaining cases the solution was prepared in the ordinary way.
For obtaining a continuous low pressure a large filter pump was used and proved satis-
factory. The mass of silver deposited per coulomb is practically the same as before and
there is, therefore, no pressure effect in the Rayleigh form of voltameter if set up and
used in the manner specified in this communication. SCHUSTER and CnosSLEYt found

TABLE VIII. — Pressure Effect. Rayleigh form of Voltameter used.

No.

Pressure
in centimetres of

Milligrammes per
coulomb.

Difference from
mean.

mercury.

19

8

l-11829tt

+ 3

20

8

25tt

-1

216

2-5

29tt

+ 3

706

2-4

28»

+ 2

756

2-4

20»

-6

Mean = 1-11826

ft Absolute determinations.

\\ Rayleigh voltameter at atmospheric pressure taken as the standard (1-11827).

that the amount of silver deposited, when their voltameter was subjected to a gaseous
pressure of about 2'8 centims. of mercury, was 4 parts in 10,000 greater than when in
air, and Dr. KAHLE| verified this result. MYERS,§ who repeated these experiments,
found the difference between deposits in air and in vacito to be as much as 1 part
in 1000, and also found an excess of 5 parts in 10,000 if the deposit was made in an

* GUTHE, 'Phys. Rev.,' 19, p. 138, 1904; ' Bull. Bureau of Stands.,' vol. 1, p. 34, 1904.
t SCHUSTER and CROSSLEY, 'Roy. Soc. Proc.,' 50, p. 344, 1892.
J KAHLE, 'Brit. Assoc. Report,' Section A, 1892.
§ MYERS, ' WIED. Ann.,' 55, p. 288, 1895.

ON THE SILVER VOLTAMETER. 569

atmosphere of nitrogen. RICHARDS and HEIMROD* verified these results. MERRILL!
varied the pressure from 1 to 103 atmospheres and found no pressure effect for this
range — he did not make olwervatious at less than atmospheric pressure.

Temperature Coefficient.

In order to investigate the effect (if any) of temperature upon the silver deposits,
we first compared the deposit in two Rayleigh forms maintained at different tempera-
tures, but in the same circuit. The lower temperature was that of the room in which
the observations were made and averaged about 16° C. The higher temperature was
that of an electric oven and was varied from 40° C. to 95° C. Sufficient distilled
water was taken to fill the voltameters and was warmed until its temperature was
comparable with that of the electric oven. The solution was made and divided into
two parts, that portion for the voltameter at the normal temperature being cooled to
16° C. and the other portion inserted in the kathode vessel of a voltameter and the
whole placed in an electric oven. The results were as follows : —

grammes. * C. grammes. " C. « —

7-00401 15 (70a) 7'00483 65 (70c) +24xlO-",

7-01085 16 (71a) 7'01210 92 (716) +24xlO~4,

7-05431 16 (72a) 7'05500 46 (726)

The values are fairly consistent, but we were not satisfied. The mean temperature
coefficient appears to be positive and about 2 or 3 parts in 1 ,000,000 per degree, but
it appeared to us that there were sufficient disturbing influences at work to account
for the higher deposits at the higher temperatures. The filter paper cup, the folds
of which were secured with platinum wire and not with wax, turned a very dark
brown colour on exposure to the atmosphere of the electric oven, and we felt that we
were not justified in using filter paper at temperatures much higher than that of the
room. The same thing happened to the exposed portions of a porous cup, and we
resolved, therefore, to use a syphon at both high and low temperatures. In addition,
we placed sheets of glass over the kathode bowl, so that, together with the limb of
the syphon which entered the vessel, they shielded the electrolyte from currents of
air. The following results were obtained : —

grammes. * C.
6-98703 16 (776)
7-00301 15 (826)

grammes. *C.
6-98754 90 (77a)
7-00342 92 (82a)

-*

+ 0,xlO

These indicate that the temperature coefficient over the range 15° C. to 92" C. is
either nil or negligibly small. Unfortunately, we could not pass a current of 1 ampere

* RICHARDS and HEIMROD, 'Proc. Am. Ac.,' 37, p. 430, 1902.
t MERRILL, 'Phys. Rev.,' 10, p. 170, 1900.
VOL. OCVII. — A. 4 D

570 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

through the electrolyte in the syphon at high temperatures. The heat produced by
the passage of such a current raised the temperature of the liquid sufficiently to
vaporise a portion of it and so break the circuit at the bend of the tube.

Our results are not in agreement with those of most other observers. Lord
RAYLEIGH* found a higher deposit at 50° C. than at 15° C., and a higher deposit at
] 5° C. than at 4° C. The temperature coefficient was therefore positive and averaged
about O'OOl per cent, per I8 C. for the range 4° C. to 50° C. LEDUC found a negative
coefficient. RICHARDS, COLLINS, and HEIMROD obtained at 60° C. and also at 0° C. a
larger deposit than at 20° C. They state, however, that the apparent gain at 0° C.
was undoubtedly due to the difficulty in washing the deposited silver. The tempera-
ture coefficient obtained from their results is about O'OOl 7 per cent, per 1° C. for the
range 20° C. to 60° C.

MERRILL, t who experimented with fused silver nitrate and solutions at normal
temperatures, concluded that temperature has no effect on the mass of the deposit.

Liquid Inchisions in the Deposit.

The appearance of the deposits in the platinum bowls varied with the solutions
used, but with pure solutions they were generally of very open texture. The
deposits were usually dried in an electric oven at 160° C., but in eight cases we
reheated at 240° C. without observing any loss in weight, and on three other occasions
some deposits in silver bowls were heated to over 400° C. In no case was there a
diminution in weight, but in two of the latter experiments a gain of 3 parts in 100,000
was recorded. This was possibly due to the formation of silver sulphide.

Lord RAYLEIGH and Mrs. SIDGWICK* sometimes found no loss on a second heating,
but more often a slight decrease. RICHARDS^ found a mean loss of 18 parts in 100.000
when the deposits were reheated over an alcohol flame to constant weight. The most
extensive observations have been made, however, by VAN DIJK,§ who after washing
and drying at 150° C. reheated in an electric furnace to 500° C., and in some cases to
600° C. No loss in weight was observed. VAN DIJK used smaller bowls and
crucibles than RICHARDS, but the amounts of silver deposited by him are comparable
with those deposited by RICHARDS ; it is apparent, therefore, that the texture of the
deposits must have been different or that something in addition to silver was
deposited in RICHARDS' experiments.

Size of Kathodes.

In general the area of the kathode surface in our experiments was 200 sq. centims.,
but in observation 58d the bowl L of 100 sq. centims. kathode surface was used, and

* RAYLEIGH and SIDGWICK, 'Phil. Trans.,' 175, p. 411, 1884.

t MERRILL, 'Phys. Rev.,' 10, p. 170, 1900.

J RICHARDS, COLLINS and HKIMROD, 'Proc. Am. Ac.,' 35, p. 145, 1899.

§ VAN DIJK, 'Ann. der Phys.,' 19, p. 266, 1906.

ON THE SILVER VOLTAMETER 571

in 58e one of the bowls (N) belonging to Professor VAN DIJK was employed. The area
of the kathode surface of this last bowl was 40 sq. centima only, and this is the smallest
k;i( bode area employed by us for currents of nominal value 1 ampere. The results of
observations 58a, I, d, and e are in remarkable agreement (Til 827, I'l 1828, 1 '1 1827,
and 1 • 11 827), and it appears that within the limits stated above the area of the kathode
has no influence on the deposit, conditionally, of course, that the concentration of the
electrolyte is within certain limits, and that the current density is not too great. In
some of our earlier experiments we used rotating kathodes ; the large platinum bowls
were rotated about 40 times per minute and stationary glass vanes were inserted in the
kathode liquid to prevent its uniform rotation. The steadiness of the current was
not appreciably affected by this motion, and had more satisfactory deposits been
obtained in the rotating lx>wls than in the stationary ones we should have no
hesitation in recommending the method. The deposits were much the same in
texture, however, and there was no difference in their masses. As an example we
may refer to 38a, 6, and c. 38a and 386 were rotated ; 38c was not. The masses of
the deposits were 7-03110, 7*03096 and 7*03108 grammes respectively.

Size of Anodes.

In most of our experiments it was impossible to estimate the extent of the
anode surface owing to the outer coating of the electrically deposited silver. The
silver plates generally used as anodes were abotit 5x5xO-4 centims., and the
current density at an anode was therefore comparatively small. In observations
25c», 26c and 416 the anodes were very small silver discs, having a total area of
about 2 '5 sq. centims. at the commencement of the deposit, and about I sq. centim.
at the conclusion ; they were not coated with electrolytic silver. On one
occasion the mass of the anode disc at the commencement was 12 grammes only,
and 7 grammes of silver were deposited. This is an extreme case. The results of
the " small anodes " observations are as follows : —

25a 1-11825.
26c 37.

416 27.

According to SCHUSTER and CHOSSLEY,* a small anode may give a deposit which
is too small. LEDUct states the opposite of this, and MERRILL^ observed no
' lillei-ence due to variation in the size of the anode. In the porous cup form GUTHE§
found the size of the anode to be immaterial, but states that the drop of potential

* SCHUSTER and CIUXSSI.F.Y, ' Roy. Soc. Proc.,' 50, p. 344, 1892.
t LKDUC, ' J. de Phys.,' 1, p. 561, 1902.
» MERRILI., ' Phys. Rev.,' 10, p. 172, 1900.
§ GUTHK, ' Bull. Bureau of Stands.,' voL 1, p. 361, 1904.

4 D 2

572 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWKY

from anode to kathode should not be large enough to allow of a decomposition of
water. We have made experiments to test this, and give the results obtained when
the potential difference was varied from 0'03 volt to 30 volts. The effect of using
a platinum anode is discussed in Part II., p. 588.

Effect of Potential Difference between Anode and Kathode (56« and 566).

When a syphon form of voltameter and a Rayleigh form were placed in series and
a current of 1 -02 ampere passed through them, the drop in potential on the syphon
form (small syphon) was of the order of 30 volts and on the Rayleigh form about 1 volt.
It has already been shown that the mass of silver deposited on these occasions is the same.
Comparison observations were also made with a very high resistance Rayleigh volta-
meter and one of the usual type. The voltameters were placed in series and the
kathode bowls of each contained 350 cub. centims. of a 15 per cent, solution of silver
nitrate. The high -resistance voltameter was one in which the filter paper was
enclosed in a glass funnel perforated with small holes, and the funnel was enclosed in
a second similarly perforated. With a current of 0'03 ampere the difference of
potential on the high-resistance voltameter was 3'0 volts, and that on the usual form
was 0'03 volt. The deposits were very loose and markedly striated. The mass of
silver deposited in 50 hours in the usual form was 7*10382 grammes (56a), and in
the high-resistance form 7 '104 11 grammes (566). These results are interpreted as
indicating that in all ordinary cases the potential difference produces no disturbing
effect.

Variable Concentration of Electrolyte.

The extreme range in the concentration of the electrolyte has been from 1*5 parts
to 50 parts of silver nitrate in 100 parts of the solution, the intermediate values being
5, 10, and 15 parts in the same quantity of solution. The chief difference in the
deposits was that of texture, the solutions of higher concentration giving less adherent
and more striated deposits than the weak solutions when the current employed was
between O'l and TO ampere, but from I'O ampere upwards the striae were faint even
for the concentrated solutions. When the 1'5 per cent, solution was used, only
3 '3 grammes of silver were present in the electrolyte, and the degree of the electrolysis
at the end of this experiment was therefore 7'l/3'3 = 2'15.

The masses of silver in milligrammes per coulomb recorded as deposited from the
various solutions are as follows : —

1-5 per cent, solution = 1-11823 (646).

5-0 „ „ = ri!8258(40a, 40c).

10-0 „ „ = T11827 (33a, 6, c).

15-0 „ „ = 1-11827 (large number of observations).

50-0 - 1-11827 (306).

ON THE SILVEB VOLTAMETER. 573

The differences from the mean value are within the limits of error, and there is,
therefore, no certain change in the mass of silver deposited per coulomb from electro-
lytes containing from T5 to 50 parts of silver nitrate in 100 parts of solution.

\\'<- tliink it necessary, however, to point out that a current of O'l ampere was used
to electrolyse the T5 per cent, solution ; when strong currents were used the silver
was deposited as long, needle-shaped crystals, and on one occasion it was precipitated
in a spongy form. It is insufficient, therefore, to state the range in the concentrations
of the electrolyte without also specifying the quantity of the electrolyte, the extent
of the kathode surface, and the current to be used.

Variation of Current Density.

We believe that one of the objections to the silver voltameter is that the ordinary
size of voltameter possibly allows only of currents of the magnitude of 1 ampere
to be measured. It was of some importance, therefore, to decide whether or not
currents of the order of half an ampere and others of the order of 10 amperes
deposited exactly the same mass of silver per coulomb in our form of the Rayleigh
voltameter. This might have been tested by evaluating the currents, noting the
times, and determining the masses, but a much simpler and more accurate way was a
comparison of the masses of silver deposited, similar to the calibration of a box of
weights. This latter method was adopted by us. In our first experiment we
compared the masses of the bowls A and- B and of C and D. B and C were then
placed in parallel and A in series with them, and a current of 1 ampere passed
through A for 2 '4 hours. A mass of silver, weighing about 10 grammes, was thus
deposited in A. The bowl D was then substituted for A and the same current passed
through it for the same time. At the conclusion of the experiment there were about
10 grammes of silver in each bowl, but the silver in A and D had been deposited
with a current of 1 ampere, and that in B and C with a current of half an ampere.
A was again compared with B, and C with D. The difference (A — B)— (C— D) should
be the same as before if the change in current had no effect. Similar observations
were made for currents in the main circuit of 2, 4, and 8 atnperea The masses of
silver deposited and the difference (A — B)— (C— D) before and after the depositions
are given in Table IX., p. 574.

We conclude from these results that the voltameter employed by us can be used
for the determination of currents as great as 8 amperes and as small as 0'5 ampere,
and that these currents will deposit the same mass of silver per coulomb. With a
current of 4 amperes the electrolyte was warm to the hand after the experiment, and
with 8 amperes the temperature rose from 18° C. to 35° C. The deposits with high
current densities are firm and of a matt surface, while with very low current densities
the silver is loose and the deposit striated.

We have made one observation which connects the deposits obtained with 1 ampere

574

Mil. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWKY
TABLE IX. — Rayleigh Form of Voltameter.

Difference in

Current

Current

Difference in

mass of bowls

Deposit in

through bowls
A and D.

through bowls
B and C.

mass of bowls
(A-B)-(C-D).

plus about 10 grammes
of silver in each

A + D greater than in
H + Cby

(A-B)-(C-D).

amperes.

amperes.

grammes.

grammes.

per cent.

1

0-5

3-13707

3-13647

-0-0030

2

1-0

3-13710

3-13676

-0-OOlr

4

2-0

3-13722

3-13788

+ 0-0033

8

4-0

3-13720

3-13720

O-OOOo

with those obtained with 075 and 0'25 ampere. In this case a resistance was added
to that portion of the branch circuit containing C for one-half of the time of the
experiment and to B for the other half. A current of 1 ampere was passed through
A and D, and through B and C currents of 075 and 0'25 ampere passed. The total
mass of silver deposited in A and D was 14 "022 15 grammes and in B and C it was
14'02246 grammes, a difference of 0'0022 per cent. (62a, b, c, d).

Effects of Electrolysis on the Concentration of the Electrolyte.

When the current first leaves the anode it spreads out in the approximately
homogeneous electrolyte which surrounds it, and, if the normal distance from the
anode to the kathode is everywhere the same, the current density over the anode
surface is uniform, and the same is true for the kathode surface. Immediately, the
layers of liquid in contact with each electrode become changed in concentration and
density : around the silver anode a film of dense liquid of high concentration is
formed and about the kathode a film results the density of which approximates to
that of water and is of very small concentration. In the Rayleigh form of voltameter
the heavy anode liquid descends, and since it constitutes a path of high conductivity,
more silver per unit area is deposited on the base of the bowl than on the sides if the
anode surface is everywhere at the same normal distance from the kathode surface.
This descending column of heavy anode liquid gives rise to the star-shaped deposit on
the base of the bowl which has been so frequently noticed by other observers and by
ourselves (fig. 8, Plate 9). If the distance of the anode from the base of the bowl is
appreciably greater than the distance from the sides, the path of least resistance is
not necessarily that of the descending column, and the deposit per unit area on the
base is less than on portions of the sides. This latter condition holds for the Rayleigh
voltameter as we have generally used it.

It follows that the lowest point of the anode is in contact with a thin layer of
electrolyte of greater concentration than the solution at the surface. A concentration
cell is thus produced and normally a current would flow through the electrolyte from

ON TIIK SILVER VOLTAMETEIi. 575

the highest point of the anode to tin- lowest. This, however, only holds good for a
few seconds after the cessation of the current in the main circuit. The siijici position
of tin- i-IVrct MM tlir main current effect results in the current density l>eing greatest
at the |>oint where the anode enters the electrolyte, and this is, in general, the first
portion of the anode which becomes noticeably thin. If the current is very feeble,
< [illusion tends to keep the liquid more homogeneous. Observations show that when
silver rods are used as anodes there are other effects of electrolysis which produce
vertical grooves in them.

If the concentration of the electrolyte is diminished, the ratio of the concentration
of the anode film to that of the main electrolyte is increased and that of the kathode
film to the liquid is probably diminished. There is, however, a similarity with respect
to the action of the current on the surfaces of separation of these films and the
electrolyte. In l>oth cases the current in its passage through the voltameter flows
from a liquid of high concentration into a mass of liquid of lower concentration, and
hence, if there is a marked surface of separation of anode liquid and electrolyte, and
kathode liquid and electrolyte, any effect of the current on these surfaces will be
similar. CORK* has shown that when a solution of small concentration rests on one
of high concentration and a current is passed downwards, the surface of separation of
the liquids becomes indistinct, but if the current is reversed the surface of separation
becomes more marked. By using "silver anodes and kathodes of platinum foil in
electrolytes contained in glass vessels we were able to see the heavy anode liquid in
its descent from the anode and the light liquid in its ascent from the kathode. Even
when at a distance of a few millimetres from the electrode these liquids appeared to
be quite distinct from the main body of the electrolyte, and we are justified therefore
in assuming the existence of even more distinct surfaces of separation around the
anode and kathode when the current is flowing.

An interesting question is whether the properties of these films of liquids are very
different from the main portion of the electrolyte. Olraervation shows that as they
leave the electrodes they break up into cylindrical columns, but whether or not they
are in the form of uniform thin films when in contact with the electrodes direct
observation does not show, but a number of experiments with currents of different
intensities aud electrolytes of different concentrations enable an opinion to lie formed.
When weak electrolytes are used (e.g., l£ per cent, solutions) a current of O'l ampere
produces a deposit having a matt surface, but with a 15 per cent, solution the deposit
is markedly striated (figs. 9t and 10, Plate 9). We interpret these results in the
following manner. The film of liquid in contact with the kathode has a greater mean
thickness in the l£ per cent, solution than in the 15 per cent, solution. This follows
because the rate of deposition of silver is the same in each voltameter. If the film is
very thin, it is unstable and breaks up into cylindrical columns of liquid. Hence in

* GORK, ' Roy. Soc. Proc.,' No. 203, p. 332, 1880, and No. 212, p. 56, 1881.

t Fig. 9 represents a portion of a deposit which was stripped from the side of a platinum bowl.

576

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

contact with the kathode surface there are columns of liquid of low concentration, and
in between these the electrolyte is of approximately normal concentration. The latter
has the higher conductivity, and since in addition there is an E.M.F. acting from the
columns of low concentration towards the main body of the electrolyte, the current
will pass into the kathode through the liquid in between the columns. Immediately
the concentration falls and possibly the resultant liquid of small density is pulled into
the columns of low concentration. If our assumptions are correct, an increase in the
current should result in the kathode film becoming thicker and more stable, and when
it is sufficiently stable to remain as a film a striated deposit should not be formed.
This was tested by experiment and found to be so.

The following table indicates the results. All the solutions were pure, and the
same volume of electrolyte (350 cub. centims.) was taken in each case.

No. of
experiment.

Electrolyte.

Current.

Character of deposit.

per cent.

amperes

1

1-5

0-1

No striae.

2

3-0

0-7

» »

3

2-0

1-0

» jj

4

15-0

0-1

Marked striae.

5

15-0

0-3

» »j

6

15-0

0-7

Striae, but not so marked as in (4).

7

15-0

1-0

Very faint striae at bend of bowl.

8

15-0

2-0

No striae.

9

15-0

4-0

j) »»

10

15-0

8-0

„ „ fine matt surface.

11

50

1-0

Striae.

It appears that for solutions of all concentrations striated deposits are obtained for
small current densities at the kathode, and matt deposits for very large current
densities.

We may now compare the changes in the Rayleigh and Richards forms of
voltameter. In the latter case there is no descending anode liquid, and there will be,
therefore, less tendency for a star-like deposit to be formed on the base of the bowl.
In our own form of RICHARDS' voltameter the volume of kathode liquid was in general
about 250 cub. centims., and during electrolysis the mean concentration of the
solution must have diminished from 15 to 10'6 per cent. The mean concentration of
the electrolyte in the Rayleigh form remains constant and, in consequence, for the
same current density striae were produced in the Rayleigh form when they were
absent in the Richards form. This effect has also been observed by GUTHE* and by
VAN DlJKf. In the Richards form, as employed by the latter observer, the kathode

* GUTHE, 'Phys. Rev.,' 19, p. 147, 1904.

t VAX DIJK, 'Ann. der Phys.,' 19, p. 271, 1906.

ON THE SILVER VOLTAMETER

577

liquid consisted of about 30 cub. centims. of a 20 per cent, solution, and at times
nearly 3 grammes of silver were deposited, the concentration being thus reduced to
about 4'3 per cent. Professor VAN DIJK. observed little or no striae in the Richards
form, but marked striae in the Rayleigh form.

In the Rayleighr form the process of intermixing of the anode and kathode liquids
is considerably accelerated by their ascent and descent respectively. We have made
observations on the currents of liquid thus produced and find that they may be
approximately represented by fig. 11. The electrolyte which is above the horizontal

Fig. 11.

plane containing the sheet of silver which forms the anode is certainly of lower
concentration than that below this plane when electrolysis has proceeded for a short
time. It therefore appears that the kathode film may be stable in the upper portion
of the electrolyte when it is unstable in the lower portion. In other words, it should
be possible to produce striae at the base and bend of the bowl when the upper portion
of the deposit presents a matt surface (see fig. 9). Probably this has been observed by
other workers besides ourselves, but we have found no reference to the phenomenon.
Our own deposits at low current densities were often markedly striated at the bend of
the bowl and gradually merged into the matt surface which existed above the point A.
The portion B of the electrolyte, which is partly enclosed by the curved surface of the
liquid, is in a most unfavourable position for the renewal of its original concentration,
and it is probably of smaller density than any other part of the solution. Because of
tliis the conductivity is small, and an appreciable E.M.F. acts from B towards the
main body of the electrolyte ; in consequence, the quantity of silver precipitated by
the passage of electricity through B is very small. In most of our experiments we
have closely observed the deposit and the electrolyte, and in about six instances have
found a distinct gap between the line of contact of electrolyte and platinum and the
edge of the deposited silver. In these cases a few crystals of silver were sometimes
VOL. ccvu. — A. 4 E

578 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

•
deposited in the form of a thin ring where the electrolyte was at its highest point, and

one or two millimetres below this ring the edge of the main deposit was formed. A
photograph of such a ring deposit is given in fig. 13 (Plate 9).

Another interesting question is whether the kathode and anode films can be easily
scattered by agitation of the liquid. We have made experiments with the kathode
only, for there were no means of detecting whether the anode film was disturbed or
not. Our solutions were such (solutions used in Observations 1 to 13) that striae
were readily produced, and the method of investigation was to rotate the kathode
and insert stationary glass vanes in the electrolyte to prevent a uniform motion. We
were astonished to find the striae as distinct as ever, but instead of being vertical they
were in the form of a spiral. This is well shown in figs. 12 and 13, which are from
photographs. Examination of the inclination shows ( 1 ) that the kathode liquid moved
upwards ; (2) that on the assumption that the kathode alone rotated and the whole
of the electrolyte remained stationary, the vertical velocity was 21 centims. per
second.* The electrolyte was not stationary, however, and we judged the kathode
film to rotate at very nearly the same rate as the kathode ; had it been at exactly the
same rate the striae would have been vertical. It is evident, therefore, that the
vertical velocity of the kathode film was not very great ; it was possibly of the
order of 1 centim. per second. The stability of the liquid columns is, however,
astonishing.

It is well to point out that the Richards voltameter is a concentration cell after
electrolysis commences. The anode liquid within the porous pot is of higher
concentration than the kathode liquid, and in consequence any short-circuiting of the
voltameter after the main circuit has been broken will diminish the mass of silver on
the kathode bowl. A steady current may thus be produced for some time, and is
easily measured by an ammeter.

Another point investigated by us was the possibility of silver being deposited from
a concentrated solution of silver nitrate at the bottom of a platinum bowl when a
second solution of much lower concentration rests on it. We employed solutions
containing 50 per cent, and 1 per cent, of silver nitrate, but neither in platinum bowls
nor in silver ones was any increase observed.

Other effects have been observed, but it is not easy to suggest an explanation of
them. The most remarkable occurs when a kathode bowl is half filled with an
electrolyte, left for an interval of about 10 minutes, and the remaining portion of the
electrolyte added just prior to the completion of the electric circuit. If about
5 grammes (or less) of silver are deposited, the level of the electrolyte when the bowl
was half filled is clearly indicated in the deposit, the density of the latter being
appreciably greater below the original level of the solution, and the change in the
density is marked by a clear line running round the bowl.

h The bowl rotated clockwise at the rate of 40 turns per minute : its maximum diameter was 10 centims.,
and the inclination of the striae was almost exactly 45° C.

ON THE SILVER VOLTAMETER. 57!)

v

'Hie Electrochemical Equivalent of Silver.

The mean of the values for the mass of silver deposited by the passage of 1 coulomb
ot electricity through any of the normal voltameters described in this communication is

1'11827 milligrammes.

The quantity of electricity which passed through a voltameter in any experiment
was determined by a measurement of time, to which measure no appreciable error can
be attached, and by an evaluation of a current in absolute measure through the
medium of the Ayrton-Jones ampere balance. It is shown elsewhere* that the error
in such a determination of current is of the order of 2 parts in 100,000, and this also
is the probable error of the value stated above for the mass of silver deposited per
coulomb.

The value has been obtained not from one solution, nor with one voltameter, but
with many solutions and many forms, as well as many voltameters. It has, moreover,
been shown that the value is the same whether a current of half an ampere is passed
through a voltameter, or a current of 8 amperes ; whether the pressure is atmospheric
or equivalent to that of a few centimetres of mercury ; and if the temperature is
90° C. instead of 15° C., it is probable that the value is still the same.

The remarkable consistency of our results is probably due to the large kathode
bowls, the purity of our anodes, the small mass of filter paper in the Rayleigh form,
and most of all to the purity of the electrolyte. With very small bowls, a small
quantity of electrolyte, a small anode, a relatively large mass of filter paper, and
current densities which are very high or very low, the estimated mass of silver
deposited in the passage of 1 coulomb may be different from the value given by us.
Secondary reactions may then occur which never happened in our experiments, or
if they did the large volume of electrolyte masked their effects and rendered their
detection impossible by any means tried by us.

Comparison of Results with those of oilier Observers.

Professor VAN DIJK very kindly forwarded to Dr. GLAZEBROOK two of his
voltameters together with a considerable quantity of silver nitrate which he had
recrystallised, and which was comparable with that employed in his investigations.
We here express our hearty thanks to Professor VAN DIJK. Solutions were prepared
from the salt which was sent, and they were found to be abnormal. In one case the
value found for the electrochemical equivalent was 1 '11847 (526), and in a second
experiment 1 '1 1860 (76/>) resulted. The solutions, as originally prepared, were slightly
turbid and had to be filtered, and the deposits with a current of 1 ampere were
noticeably striated. If different quantities of an abnormal electrolyte are contained in

* AYRTON, MATHER, and SMITH, 'Phil. Trans.,' A, vol. 207, p. 534, 1908.

4 E 2

580

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

two similar voltameters in series, the one containing the least quantity of solution does
in general give the smaller deposit. This is well illustrated in 77 'c and d. In one of
these, 450 cub. centims. of an abnormal electrolyte gave 1 '12007 as the value of the
electrochemical equivalent, while the other, which contained 150 cub. centims., gave
I'll 923. With an abnormal electrolyte a large bowl will therefore give a heavier
deposit than a small one, but although there is an appreciable difference in the size of
the Rayleigh bowl and the Richards crucible which Professor VAN DIJK used, we do
not think that the differences which he observed are to be entirely attributed to this
cause.

Professor WATSON also sent a silver nitrate solution used by him in 1895. This
gave I'll 872 (23a) for the electrochemical equivalent, and when diluted to a 7£ per
cent, solution it gave 1'11850 (60a) with a current of 2'3 amperes, and 1'11837 (61a)
on a second electrolysis with a current of O'l ampere.

In view of these facts and the great difference in the size of the voltameters, &c.,
used by other observers, it appears that no very useful purpose would be served by
an attempt to explain the results of other experimenters without first reproducing as
nearly as possible the conditions under which they worked. This we shall endeavour
to do.

It is of some interest, however, to compare the results obtained by absolute
methods, since a form of voltameter practically identical with that used by Lord

TABLE X.

Observer.

Year.

Value.

MASCART*

1884

milligrammes per coulomb
1-1156

FR. and W. KoHLRAUSCHf . .
RAYLEIGH and SmcwicKj . .
PELLAT and POTIER§ ....
KAHLE||
PATTERSON and GUTHEH . . .
PELLAT and LEDUC** ....
VAN DIJK and KuNSTft . . .

GuTHEjf

1884
1884
1890
1899
1898
1903
1904
1906

1-1183
1-1179
1-1192
1-1183
1-1192
1-1195
1-1182
1-1182

* MASCART, ' J. de Phys.,' 3, p. 283, 1884.
t FR. and W. KOHLBAUSCH, ' WIED. Ann.,' 27, p. 1, 1886.
\ RAYLEIGH and SIDGWICK, 'Phil Trans.,' 175, p. 411, 1884.
§ PELLAT and POTIER, ' J. de Phys.,' 9, p. 381, 1890.
|| KAHLE, 'Zeitschr. Inst.,' 18, pp. 229-267, 1898.
U PATTERSON and GUTHF, ' Phys. Rev.,' 7, p. 257, 1898.
** PELLAT and LEDUC, 'C. R.,' 136, 1649, 1903.

ft VAN DIJK and KUNST, 'Ann. der Phys.,' 14, p. 569, 1904. VAN DIJK, 'Ann. der Phys.,' 19,
p. 249, 1906.

\\ GUTHE, ' Bull. Bureau of Stands.,' vol. 1, No. 1, p. 36, 1904, and vol. 2, p. 70, 1906.

ON THE SILVER VOLTAMET1!! 581

I! vi i .IK. ii li;is hfcii employed on most occasiona We are unaware of the exact
conditions of the experiments, but it will be seen that the differences between the
values and the mean value are not appreciably greater than the probable errors of
many of the current determinations. It must be remembered that in the very early
observations great precision was not claimed.

PATTERSON and GUTHE used a type of voltameter in which the electrolyte was
saturated with silver oxide, and their result is not therefore comparable with the
others. (JiTHK and VAN DlJK employed the Richards and the Kuyleigb forms; the
values given in Table X. are those obtained by them for the Rayleigh pattern.

Conclusions.

(1) The Rayleigh, the Richards, the Syphon, the Pot-Syphon-Bowl, the Syphon-
Pot-Bowl, and the Elevated Kathode forms of voltameter give identical values within
1 or 2 parts in 100,000 for the electrochemical equivalent of silver, subject to easily
attained conditions with respect to the size of the voltameter and the purity of the
electrolyte. (The purity of the electrolyte is dealt with in Part II.)

(2) The mass of silver deposited is independent of the pressure to which the
voltameter is subjected, and also independent of the temperature, except that at high
temperatures the filter paper of the Rayleigh form may interact with the silver
nitrate solution, and give rise to a very slightly abnormal value for the electrochemical
equivalent of silver.

(3) The current through the Rayleigh form ot voltameter* may vary from
0'5 ampere to. 8 amperes, and possibly beyond these limits without producing any
appreciable disturbing effect.

(4) The electrochemical equivalent of silver is

1'11827 milligrammes of silver per coulomb (10"1 C.G.S.).
This value is subject to a probable error of about 0'002 per cent.

PART II.

The Chemistry of the Silver Voltameter ;
by F. E. SMITH, A.R.C.Sc., and T. M. LOWRY, D.Sc.

A. Preparation of Pure Silver Nitrate.

In the earlier experiments (1 to 13) on the electrochemical equivalent of silver con-
siderable difficulty was experienced in obtaining concordant figures when different
samples of silver nitrate were electrolysed under apparently identical conditions. Many

* The size of the voltameter is assumed to be the same as that described in this communication.

582 MR. F. E. SMITH, MR, T. MATHER, AND DR. T. M. LOWRY

of the solutions used had been prepared from silver nitrate recovered from previous
electrolyses, purified by adding nitric acid, boiling down to dryness, and fusing in a
platinum basin ; the fused mass was dissolved in water, filtered from the black residue
which was always left after fusion, and was often used for electrolyses without further
purification. On a few occasions silver nitrate was crystallised from the filtrate and
a 15 per cent, solution made from this recovered salt. The values for the electro-
chemical equivalent of silver varied from T11832 to 1'11886 milligrammes per
coulomb (mean of 14 determinations = 1 "11857) when the Rayleigh voltameter was
used, and from T11786 to T11854 (mean of 21 determinations = T11825) when the
porous pot voltameter was used.

Before a definite figure could be established for either form of voltameter it was
necessary first to demonstrate the possibility of preparing again and again from silver
nitrate of different origins solutions which should give identical weights of silver
when electrolysed under identical conditions. The following experiments were there-
fore made in order to test the constancy of the electrochemical equivalent of a range
of silver nitrate samples of different origins.

I. Silver Nitrate from Electrolytic Silver. — 185 grammes of electrolytic silver
recovered from previous electrolyses were dissolved in a mixture of equal volumes
of " commercial pure " nitric acid and water. The resulting solution was filtered, by
means of a small Gooch crucible, from a small residue of insoluble matter,* and
evaporated on a water-bath (since it was not thought to be desirable to fuse the
product) during 50 hours, water being added from time to time. A crop of crystals
was then drained off and dried overnight in the oven. When recrystallised the
product was found to be neutral, but yellow in colour. After three further recrystal-
lisations the electrochemical equivalent of the sample was tested. The solution for
electrolysis was prepared by dissolving 90 grammes of the purified salt in 600 cub.
centims. of water of low conductivity, and when electrolysed in a Kayleigh voltameter
gave a deposit of I'll 825 milligrammes per coulomb (246).

II. Prom Recovered Silver Nitrate. — 850 grammes of strongly acid crystals,
recovered from previous electrolyses by acidifying and concentrating the solutions,
were purified by repeated crystallisation only, without any attempt to remove the
acid by evaporating on the water-bath, by drying in an oven, or by fusing. After
four crystallisations the nitrate was found to be neutral to litmus, and after one
further crystallisation it was dissolved in water (113 grammes AgNO3 in 750 cub.
centims. of water of low conductivity) and electrolysed. The electrochemical
equivalent of the sample was found to be 1 '11827 milligrammes per coulomb (24a).

h The insoluble residue referred to was suspended in very minute particles in the liquid, to which it
imparted a red colour suggestive of a colloidal metal. It was found to be platinum, and 0-0136 gramme
was obtained from 185 grammes of electrolytic silver. The impurity, therefore, was probably not a
product of electrolysis, but may have been derived from the platinum bowls or from the platinum spatula
which was used to remove the electrolytic silver.

ON THE SILVER VOLTAMKTF.K 583

III. The strongly acid mother-liquors from II. were evaporated on the water-bath
and dried in the oven until free from acid. The dry salt was further lu-.itcd until thr
hlue colour of the copper nitrate had disappeared, and was then dissolved in water,
filtered, and recrystallised until colourless. After one further crystallisation a
solution was prepared as before, and the electrochemical equivalent determined.
Three determinations gave the values

III. 1-11827 (266)
1-11830 (27 d)
1-11827 (29a)

Mean 1*11828 milligrammes per coulomb.
Other samples of recovered silver nitrate purified by this method gave the values

IV. 1-11826 (28c)
V. 1-11831 (356)

1-11827 (35c)

VI. 1-11816 (?) (44c)

1-11830 (46c)

VII. 1-11830 (656)

whilst a 50 per cent, solution prepared by dissolving recovered silver nitrate (purified
as above) in its own weight of water gave the value

VIII. 1-11827 (306).

IV. Silver Nitrate from Commercial Samples. — As a further test of the possibility
of preparing silver nitrate of constant properties, two commercial samples were
procured, one from an English and one from a German firm, and these were purified
by recrystallising two or three times from water. One of these samples was of
especial interest, as it gave initially a lower value for the electrochemical equiva-
lent than any of the samples I. to VIII. which we had purified for ourselves, and
we were uncertain at first whether a higher standard of purity might not perhaps
be attained by a works-recrystallisation on a large scale than when smaller quantities
were dealt with. The recrystallised samples gave, however, entirely normal values,
and the low value of one of the commercial samples was therefore due to an
impurity which could be removed by recrystallising (see later p. 595). The values
obtained for the electrochemical equivalent were : —

IX. T11830

(26a)

1-11828

(27c)

1-11828

(286)

X. 1-11827

(44a)

XI. 1-11826

(446)

584 MR F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

It will be seen that with one exception all the eleven solutions gave, when
electrolysed under similar conditions, values for the electrochemical equivalent lying
between 1 '11826 and 1 '11831, the mean value being T11828 milligrammes per
coulomb. The average error for the nine concordant solutions is 0 '00001, and is of
about the same magnitude as that observed in the case of duplicate determinations
with the same sample of nitrate.

The only discordant value in the above series (No. VI.) gave on a first electrolysis
the figure I'll 816, but this may possibly have been due to some accident of manipula-
tion, e.g., to the loss of a trace of loose silver from the bowl ; a redetermination of the
electrochemical equivalent gave the value I'll 830, a figure which differs only by
0 '00003 from the mean. One solution prepared subsequently for use in a com-
parative test gave the value 1'11836 (52c), but the nitrate used for making the
solution had been exposed to the air for a long time and was considered not to be
sufficiently pure for an absolute determination.

B. Tests of Commercial Silver Nitrate.

Having established a definite figure for the weight of deposit obtained from highly
purified samples of silver nitrate when electrolysed under standard conditions, it was
desirable to ascertain how far commercial " pure " silver nitrate could be relied upon
to give a correct weight of deposit. The result was encouraging in so far as with
one exception all the samples examined gave figures agreeing with those obtained
in the preceding section A. The values for the different samples were as follows :—

Hj I'll 827^ Very large number of

H2 1'11827 /» observations (see

H3 1-11827 J Table I.).

Wt T11819 (226, 236, 25a, 256, 316).

G! 1-11827 (30a).

M T11829 (30d).

G2 1-11827 (38a).

W2 1-11826 (48a).

The abnormal specimen W\ gave a normal deposit (IX.) when recrystallised and
was therefore considered to contain a removable impurity. When the 15 per cent,
solution was tested with neutral litmus paper it did not show any acid reaction ; but
a more concentrated solution tested with blue litmus showed a marked acidity which
was absent from the solutions which gave normal deposits. The acidity of the
specimen was further established by precipitating the solution with neutral sodium
chloride and testing with methyl orange. The makers subsequently stated that the
nitrate had been crystallised from a slightly acid liquid in order to secure the forma-
tion of clear crystals, and there can be little doubt that a trace of acid mother-liquor

ON THE SILVER VOLTAMETER. 585

had been retained, and that this was the cause of the abnormal character of the

ilr|i"sit. Tin1 Iliakrix \\n\\ anli-d ;i v.-,-,,::,! v;llli|i]r \\!il<-!i MM <TV>-talli-i-il t'p-Ill a

neutral liquid, and normal results (W,) were obtained with this.

C. Standard Method of Preparing Silver Nitrate Solutions for Electrolysis.

It is now possible to state the conditions that 'should be complied with in preparing
silver nitrate solutions for use in the voltameter.

(1) If commercial " pure " silver nitrate is used, a part of it should be purified by
recrystallising twice from water and the deposit compared with that from the
original sample. If the values agree, this can be used without purification ; if not, the
\\hole of the sample should be twice recrystallised. For rough work in which an
error less than O'l per cent, may be neglected the commercial nitrate may be used
directly without testing or purifying.

(2) If recovered silver nitrate is used it should be freed from acid by evaporating to
dryness and heating in the oven at 140° C. until the blue colour of the copper nitrate
(if present) is destroyed, then dissolved, filtered and recrystallised until the mother-
liquor drained from the crystals is colourless, then once again recrystallised before
being used for electrolysis.

(3) In crystallising the nitrate it is desirable to effect the dissolution of the crystals
by heating on a water-bath, rather than over a bare flame, so as to avoid all risk of
overheating the solution, and to dissolve in a conical flask of Jena glass rather than in
a beaker, so as to reduce the risk of contamination by exposure to the air.

(4) For filtering the hot solution we prefer to use a Hirsch porcelain funnel, the
perforated plate of which is covered by two discs of filter paper. The funnel is
attached to a filter pump and warmed by pouring lx>iling distilled water through it ;
there is then but little risk that the hot solution will crystallise in the filter. The
filtered solution is allowed to crystallise in the pump-flask, so as to avoid unnecessary
exposure to the air. If the flask is cautiously cooled and shaken it is generally
possible to secure the separation of the nitrate in small crystals ; these can subse-
quently be drained much more effectively than the larger crystals, which separate
when the solution is allowed to cool slowly and without disturbance. As an alterna-
tive the filtered solution may be left to deposit large crystals from which the mother-
liquor can be poured off, but in this case the separation of the mother-liquor is much
less complete and a larger number of crystallisations is required. Towards the
end of the crystallisation the flask may be cooled in ice, so as to reduce the amount
of material left behind in the mother-liquors.* A porcelain filter funnel may be used
without filter paper for collecting and draining the crystals ; when these are well
pressed down in the funnel, most of the mother-liquor can be removed by means of a

* 100 grammes of water dissolve 115 grammes AgNOj at 0°, 160 grammes at 10*, and 215 grammes
at 20° C.

VOL. CCVII. — A. 4 F

58« MR. F. E. SMITH, MR. T. MATHER. AND DR. T. M. LOWRY

filter pump, but if the liquors contain much impurity it is advisable to rinse the
crystals cautiously with a few cubic centimetres of iced distilled water.

(5) If it is desired to dry the crystals, the best method is to make use of a
HEMPEL'S vacuum desiccator charged with stick potash, and cautiously heated over a
water-bath to accelerate the drying. As a rule, however, it is best to use the moist
crystals for preparing solutions, the exact strength of which can, if necessary, be
determined by evaporating a known weight of the solution.

(6) The water used in the earlier experiments for the final crystallisation of the
nitrate and for the preparation of the solutions was a specially pure sample prepared
at Hendon by Mr. W. R. BOUSFIELD, by a process of continuous fractional distilla-
tion, and stored in a large Welsbach bottle. Its electrical conductivity had been
measured and found to be only 1 reciprocal megohm per centimetre cube. Subsequent
experiments showed that commercial distilled water could generally be used without
introducing any error.

D. Effects produced by Repeated Electrolysis.

That an increase in the value obtained for the electrochemical equivalent of silver
may result from repeated electrolysis of a silver nitrate solution was first observed by
NOVAK,* and later by RODGER and WATSON,! KAHLE,^: VAN DIJK§ and GUTHE.||
RODGER and WATSON record as successive relative values the numbers

9983, 9987, 9990, 9999, 9995, 9993, 9995, 10002, 10005, 10006, 10002.

In an attempt to confirm these observations we repeatedly electrolysed two solutions
of silver nitrate, the one being contained in a platinum bowl and the other in a silver
bowl. The resulting values of the electrochemical equivalent were,

when the platinum kathode was used : — and when a silver kathode was used :—

1-11827 (14a) 1-11825 (146)

1-11822 (156) 1-11826 (15a)

1-11822 (166) 1-11825 (16a)

1-11827 (17 c) 1-11833 (176)

1-11838 (18a) 1-11840 (186)

1-11834 (21a) 1-11830 (21c)

It thus appears that there may be a small increase, but nothing comparable with that
observed by RODGER and WATSON.

* NOVAK, ' Proc. Roy. Bohemian Ac. Sci. Prague,' 1, pp. 387-432, 1892.

t RODGER and WATSON, 'Phil. Trans.,' A, 186, p. 631, 1895.

{ KAHLE, 'Zeitschr. lust.,' 18, pp. 229-267, 1898.

§ VAX DiJKand KUNST, ' Ann. der Phys.,' 14, p. 569, 1904. VANDlJK, « Ann. derPhys.,' 19, p. 249, 1906.

|| GUTHE, ' Phys. Rev.,' 19, p. 138, 1904 ; ' Bull. Bureau of Stands.,' vol. 1, p. 355, 1904.

ON THE SILVER VOLTAMETER. 587

In a final experiment a current of 1 ampere was passed through a solution during
200 hours until the solution was hlue with copper from the " pure " silver anode.
During this interval the mass of silver transferred through the solution amounted to
MM It-ss than 820 grammes, or more than 100 times as much as in a normal
electrolysis. The values obtained for the electrochemical equivalent were : —

Before electrolysis .... 1 '11827 (solution H,)

After 100 hours 1-11822 (17a)

After 200 hours 1 '11832 (23d)

We were therefore driven to the conclusion that the mere act of electrolysis is not
sufficient to produce the effects recorded by RODGER and WATSON and others.

The only change which we have been able to detect in the properties of the solution
after repeated electrolysis is that it gives somewhat more coherent and very slightly
striated deposits. KAHLE* has pointed out that the filter paper may have an effect
on the electrolyte, and this view is adopted by VAN DiJKf in explanation of the
increase which he observed. In our experiments the volume of the electrolyte was
exceptionally large, and the mass of filter paper J immersed was small ; contamination
of the solution by the paper would therefore not be likely to produce any marked
effect in the course of a few electrolyses at ordinary temperatures ; at higher
temperatures we have reason to think that important effects may be produced by the
action of the filter paper on the electrolyte. No filter paper was used in the 200-hour
electrolysis except during the actual determination of the electrochemical equivalent.

E. The Question of Anodic Impurities.

It has been generally assumed that the high values obtained on repeated electrolysis
were due to the introduction of impurities at the anode during electrolysis. It was
to overcome the supposed effects of such impurities that the porous pot voltameter
was introduced by RICHARDS. We were at first inclined to agree with this view ; it
is well known that when silver nitrate is electrolysed with a platinum anode, crystals
are formed of a " peroxynitrate," Ag7NO,,. This substance was discovered by RITTER§
in 1804 ; its composition has been established by repeated analysis,]] and it is known
to be decomposed when warmed with water at temperatures from 25° upwards
according to the equation

AgTNOn = AgNOa+SAgA+O,,

» KAHI.R, 'Zeitechr. Instr.,' 22, p. 155, 1902.
t VAN DIJK, 'Ann. der Phys.,' 19, p. 249, 1906.
J SCHI.KICHKK and SCHULI., No. 595.

§ HITTER, ' GERLEUS Neuea J.,' 3, p. 561, 1804. SULC, ' Zeitechr. Anorg. Chem.,' 12, p. 90, 1896.
i| MULDER and HERINT.A, ' Verb. Ron. Ak Wet.,' 3, p. 37, 1896. TANATAR, ' Zeitechr. Anorg. Chem.,'
28, p. 331, 1901 WATSON, 'Trans. Chem. Soc.,' 89, p. 578, 1906.

4*8

588

MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

and this affords the only general method of preparing silver peroxide. It was
possible that whilst this compound does not crystallise out when a silver anode is
used, and does not occur in the anode slime, it might be produced in small quantities
and pass into solution, especially if high current densities are used at the anode. It
was found, however, that no increase in the electrochemical equivalent resulted when
the area of the silver anode of the Rayleigh voltameter was made very small and when
high current densities were employed (pp. 571 and 573, Part I.).

In further experiments it was shown that this action at the anode, which is
accompanied by a liberation of acid, actually lowers the value of the electrochemical
equivalent instead of raising it. An apparatus was arranged with a platinum crucible
as an anode, and between it and the cathode a large filter paper cup was suspended
to prevent any crystals of Ag7NOn falling on the platinum bowl. The solution was
very acid after electrolysis, and the resulting values of the electro-chemical equivalent
were 1'11779 (34a) and 1'llSll (55&). The abnormally high equivalents are therefore
not due to the formation of peroxynitrate at the anode.

RICHABDS* found that the anode liquid was so changed during electrolysis that it
deposited silver on prolonged contact with silver crystals. We have been unable to
confirm this in our voltameters.

The following experiment was performed so that the anode liquid should come
into contact with the silver surface a few seconds after its formation. The anode
and kathode liquids were contained in two silver bowls (fig. 14) connected by a

Fig. 14.

syphon, and the anode was a silver plate which dipped into a glass funnel fitted with
a filter paper. On electrolysis, the dense liquid descended to the bottom of the bowl
and thus came into contact with silver. The anode bowl was weighed both before
and after electrolysis, but no gain in weight was recorded in any of the experiments,
even though on one occasion 20 grammes of silver were deposited on the kathode
bowl. In the first experiment the silver plate was surrounded with filter paper only,
but this led to complications, owing to part of the current entering and leaving the

* RICHARDS and HEIMROD, 'Proc. Am. Ac.,' 37, p. 431, 1902.

ON THE SILVER VOLTAMETER 589

bowl surrounding it. The glass funnel largely prevented such a distribution of
current.

The experiments recorded in Part I. indicate further that, under normal conditions
of working, the exclusion from the kathode vessel of the anode liquid by means of a
porous pot is without influence on the deposit, and we therefore conclude that the
cause of the abnormal values in our voltameter (Observations 1 to 13) is to be sought
in the contamination of the solution, but that this is not due to any change which is
inseparably connected with the conditions of electrolysis.

F. Examination of the Mother- Liquors from Recovered Silver Nitrate.

Although it was not found possible to obtain appreciably higher values for the
electrochemical equivalent by repeated or prolonged electrolysis, or by bringing the
anode liquid into contact with the kathode, it was known that high values could
readily be obtained by using recovered silver nitrate that had been cleared with acid
and rendered neutral by fusion, but not otherwise purified. It was therefore thought
to be desirable to investigate these solutions in order to determine the nature of the
impurity which they contained. For this purpose the mother-liquors left behind
during the purification of the recovered nitrate, as described in § A II. and III., were
collected and examined. After recovering in a pure state the greater part of the
850 grammes of nitrate there remained a yellowish liquid containing about half its
weight of silver nitrate. On dilution with water the liquid became turbid and a
thick brown cloud was formed which ultimately settled down as a black precipitate.*
The diluted solution, which contained 14'4 per cent. AgNOa, was then electrolysed
and gave the extraordinary value

T12141 milligrammes per coulomb (27b)

for the electrochemical equivalent. A second electrolysis gave the value

1-12055 (30c),

and a third electrolysis, after the addition of 1 gramme of crystallised ferric nitrate to
about 400 cub. centims. of the mother-liquor, gave the value

1-12171 (74d).

As it appeared that the impurities were largely precipitated by diluting to 15 per
cent., an electrolysis was carried out with a more concentrated mother-liquor con-
taining 43 per cent. AgNO3. The value obtained for the electrochemical equivalent
was 1 '12252 (476), which is no less than 0'00425, or 0'36 per cent., higher than the
normal, and is possibly the highest yet recorded.

* A slight cloudiness had already been noticed when the fused nitrate referred to at the beginning of
the section was dissolved in water and diluted after filtration instead of before.

590 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

The impurity present in the mother-liquor evidently consists, then, to a considerable
extent of substances which are insoluble in water, but soluble in silver nitrate solu-
tions of moderate concentration. These impurities are partially precipitated on
diluting the solutions, but a certain amount remains in solution. The property of
dissolving in silver nitrate appears to be possessed by a large range of silver salts
which are insoluble in water, and many of these possess the property of increasing the
weight of the deposit obtained on electrolysing the solution. The removal of silver at
the kathode leads, as is well known, to the formation of a film of dilute solution which
(owing to its small density) flows upwards towards the surface of the solution. The
film immediately in contact with the kathode may be regarded as almost pure water,
and the dilution from 15 per cent, to nearly 0 per cent, (by removal of silver nitrate
instead of by addition of water) probably causes a precipitation of impurity similar to
that which results from the dilution from 50 per cent, to 15 per cent. In this
direction we believe an explanation may be sought of the abnormally high equivalents
obtained from impure silver nitrate solutions.

In the course of the later work several centigrammes of the precipitate formed on
diluting the silver nitrate mother-liquors were collected and fractionated as

follows : —

gramme.

Soluble in dilute nitric acid (Ag2O, &c.) . . 0'0231

„ ammonia (AgCl, &c.) 0'0664

Undissolved (Ag2S, &c.) 0-0274

0-1169

From the first fraction there was recovered 0'0113 gramme AgCl, equivalent to
0-0091 Ag20 ; from the second fraction 0*0567 gramme AgCl. It appears, therefore,
that about half the residue consisted of silver chloride, whilst the remaining part
contained sufficient oxide and sulphide to give the precipitate a black colour, and so
disguise to a large extent the chloride which was its chief constituent.

G. Stnation of the Deposit.

It was noticed very early in the investigation that high values for the electro-
chemical equivalent were almost invariably accompanied by a characteristic striation
of the deposit, whilst normal equivalents were almost always obtained from unstriated
deposits with currents of 1 ampere ; it was in fact possible to guess roughly what the
weight of the deposit would be by noting the appearance of the silver deposited in
the bowl. The impurity which causes the high values is evidently characterised by
the property of producing marked striations, and this property was for some time the
only qualitative test for the presence or absence of the substance in the silver
solutions. It was also considered to be of importance in seeking to determine the

ON TIII-: sii.vr.i: YOI.TAMKTI i:

ii it me «»f the impurity ; thus if high values could be obtained with a variety of added
impurities, that one which most readily gave striated deposits was the most likely to
be the characteristic impurity of the actual solutions used for electrolysis.

The deposits from the mother-liquors showed an exceedingly marked striatiou,
although the silver was dull in appearance.

H. Influence of Oxide, Carbonate and Chloride.

The influence of silver oxide on the silver voltameter has been investigated by
PATTERSON and GUTHE,* GUTHE, t RICHARDS^ and KAHLE§. PATTERSON and GUTIII
used a solution saturated with silver oxide, and GUTHE'S comparison of it with the
Rayleigh type showed the two to agree. RICHARDS, however, found his form of
voltameter to give a deposit lower by 0*1 per cent, when compared with PATTERSON
and GUTHE'S type, and KAHLE§ found the effect of silver oxide was to increase the
deposit by 0'05 per cent. From the point of view to which we have referred it was
probable that any silver salt which was insoluble in water would, if dissolved in the
nitrate solution, give an abnormally heavy deposit. Two experiments made with
solutions containing silver oxide gave confirmation to this view. Pure sodium
hydroxide was prepared by the action of water-vapour on metallic sodium and was
added to a 50 per cent, solution of silver nitrate ; the filtered solution gave a
slight brown precipitate on diluting to 400 cub. centims., and on electrolysis it
gave for the electrochemical equivalent the value

1-11852 (50rf),

0'021 per cent, higher than the normal figure. A later experiment carried out in the

same way gave the value

1-11842 (80c).

These results are not directly comparable with those of other observers owing to the
difference in the size of the voltameters, but may be regarded as substantially in
agreement with that of KAHLE.

Very similar results were obtained on adding sodium carbonate, which raised the
electrochemical equivalent by about two parts in 10,000 (Observation 536).

Addition of potassium chloride gave the values

1-11840 (50c), 1-11847 (806).

The chloride is freely soluble in concentrated silver nitrate solutions, especially when
hot, and is copiously precipitated on dilution ; its effect on the electrochemical

* PATTERSON and GUTHE, ' Phys. Rev.,' 7, p. 257, 1898.
t GUTHK, 'Phys. Rev.,' 19, p. 145, 1904.
J RICHARDS and UKIMROD, 'Proc. Am. Ac.,' 37, p. 426, 1902.
§ KAHLE, ' Brit. Assoc. Report,' Section A, 1892.

*592 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

equivalent is, however, unimportant, possibly because a sufficient weight is not
retained by the 15 per cent, solution.

The above results go far to justify the view that the majority of silver salts which
are insoluble in water dissolve to a slight extent in concentrated silver nitrate solu-
tions, and that the increase in the electrochemical equivalent which usually results
may be related to this difference in solubility, which probably acts by causing a
precipitation of the sparingly soluble salt from the impoverished solution at the

kathode.

I. Influence of Sulphide.

In view ot the readiness with which metallic silver blackens on exposure to air, it
was evident that silver sulphide was likely to be a frequent impurity in the nitrate
solutions. The blackening of the bottles in which silver nitrate solutions are kept is
a universal experience in the laboratory, and in voltameter work the blackening of
the porous pots has constantly proved a source of trouble. The sediment from the
silver nitrate mother-liquors undoubtedly contained sulphide, and it was therefore
very important to determine the influence of this substance on the electrochemical
equivalent.

(1) A solution of 60 grammes of silver nitrate in an equal weight of water was
prepared in a wide test-tube, and a small volume of hydrogen sulphide gas was
delivered into the tube above the surface of the solution. The first effect of the
sulphuretted hydrogen was to produce on the surface of the solution a yellow film
which turned black where the gas was present in largest quantities. On shaking the
solution a granular precipitate of a canary -yellow colour became distributed throughout
the solution, and the black sulphide disappeared. It was evident that where the
nitrate was in excess the precipitate was stable in a yellow form, and became black
only when the proportion of sulphuretted hydrogen to nitrate was increased. When
a considerable quantity of the yellow precipitate had been formed the solution was
filtered. On diluting with distilled water to 400 cub. centims., a brown cloud
appeared which slowly settled to a black precipitate at the bottom of the colourless
solution. The behaviour of the sulphide solution was thus essentially similar to that
of the silver nitrate mother-liquors. The diluted solution was filtered and transferred
to a voltameter. It gave a normal electrochemical equivalent

I'll 828 milligrammes per coulomb (87 6).

Three other solutions (856, 85c, 86c), prepared by somewhat similar methods, also

gave normal values,

1-11829, 1-11828, 1-11828.

We are therefore driven to the conclusion that although many of the heavy
deposits were obtained from solutions which were undoubtedly contaminated with
sulphide (mother-liquors 276, 476), the presence of this substance alone is not sufficient
to account for the production of these abnormally high values.

ON THE SILVER VOLTAMETER. 593

< )ur experiments have shown that the abnormally heavy deposits cannot be explained
HH clue to sulphide acting in presence of copper or of iron.

Two solutions were prepared by dissolving 60 grammes of silver nitrate in
60 grammes of water, and to each of these was added 0'12 gramme of commercial
pure copper foil ; this was left in contact with the solution until the copper had
dissolved as nitrate by displacing an equivalent quantity of metallic silver. Hydrogen
sulphide was then added and the solution filtered and diluted as before ; very little
sulphide was precipitated by diluting, and the electrochemical equivalent was found
to be in the case of the first solution

1-11850 (67c);

in the case of the second solution, to which hydrogen sulphide had been lavishly
added, a nearly normal value was obtained,

1-11824* (74a).

Similar experiments were made with iron. This could not be introduced in the
same way as the copper, for the metal appeared to become passive in contact with the
strong nitrate solution and refused to dissolve. Two solutions were prepared by
adding 1 gramme of crystallised ferric nitrate to GO grammes of silver and adding
sulphuretted hydrogen as before. The first solution gave the perfectly normal electro-
chemical equivalent

1-11825 (67d),

and the second, to which much more hydrogen sulphide was added, gave the value

1-11834 (74c),

also substantially normal. The ferric nitrate was strongly acid, and this fact must
be taken into account in discussing the above result, but it is clear that the
extraordinarily high electrochemical equivalent of the mother-liquors cannot be
attributed to the presence either of iron or of copper. Addition of ferric nitrate to
the mother-liquor produced no marked change in the electrochemical equivalent, the
value obtained being 1-12141 (276) and 1 '12055 (30c) before and 1 '121 71 (74</) after
the addition of 1 gramme of ferric nitrate to about 400 cub. centime, of 15 per cent,
mother-liquor.

J. Influence of Nitrite and Hyponitrite.

The abnormally high deposits obtained with the silver voltameter have usually
been attributed to anodic impurities. Such impurities would normally be oxidised
substances comparable with the persulphuric acids, with lead peroxide, or with silver
peroxynitrate. RICHARDS has, however, made the suggestion that reduction may

* The fact that this figure is somewhat lower than the normal may be due to the trace of acid which
is liberated by the sulphide 2AgNOs + SHj = AgjS + 2HN08. The solution did not, however, appear acid
to litmus.

VOL. CCVII. — A. 4 O

594 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

take place at the anode with formation of silver nitrite. It was therefore of interest
to determine the effect of this substance on the electrochemical equivalent.

RICHARDS* prepared silver nitrite by boiling silver nitrate solution with finely
divided silver, and obtained with a nitrate solution saturated with it a value identical
within 1 part in 200,000 with that found with a pure nitrate solution. He also
prepared nitrite from pure potassium nitrite and silver nitrate and concluded, from
voltameter experiments made with a nitrate solution saturated with the salt so
prepared, that the nitrite caused an increase in the deposit of between 30 and 80
parts in 100,000.

A first experiment, in which a solution of pure silver nitrate was saturated with
silver nitrite (purchased as pure) by making a saturated solution of the latter and
dissolving the requisite quantity of silver nitrate in it, gave T11832 (2id) as the
equivalent. That the solution was saturated with nitrite was evident from the
copious precipitate of this salt which resulted when the silver nitrate crystals were
dissolved in it.

A second experiment, in which a 50 per cent, solution of silver nitrate was saturated
with nitrite by the addition of KNO2 and then diluted and filtered, gave the value
1 '11 837 (52d) for the electrochemical equivalent. In this case the saturation of the
solution was evidenced by the fact that needles of silver nitrite actually crystallised
out from the solution on standing in a cool place.

The above experiments show clearly that silver nitrite even when present in
considerable quantity does not raise the electrochemical equivalent by more than
one part in 10,000 and can only be an unimportant impurity. In view of the yellow
tint of the nitrite crystals, which should be colourless when pure, we are by no means
certain that the slight increase which we have observed may not be due to hyponitrite
(vide infra) and not to the nitrite itself.

The influence of hyponitrite on the deposit does not appear to have been determined.
The salt was prepared by reducing sodium nitrite with sodium amalgam, neutralising
with acetic acid and precipitating with dilute silver nitrate solution. The yellow
precipitate of silver hyponitrite was drained on a filter, thoroughly washed with
water and shaken up with 120 grammes of a 50 per cent, solution of silver nitrate.
The strong saturated solution was then filtered and diluted to 400 cub. centims. with
water. A slight cloud was produced which was removed by filtration and the
solution was then electrolysed. The deposit was strongly striated and gave the value
1'11873 (88b). An earlier preparation gave a higher figure, 1'11930 (47c), but we do
not wish to lay stress on this.

K. Influence of Acid*.

It has been shown that there are a considerable number of impurities which raise
the electrochemical equivalent of a silver nitrate solution — oxide, carbonate, chloride

* RICHARDS and HEIMROD, ' Proc. Am. Ac.,' 37, p. 423, 1902.

ON THE SILVER VOLTA.M! II l: 595

i in I nitrite increasing the value by about one part in 10,000 and hyponitrite by about
one part in -_'<K)0. Occasionally, however, abnormally low values are obtained, as, for
instance, in the case of the commercial sample W,, which gave the mean value
1"11818, and in a large number of the earlier porous pot exjxiriments 1 '11786 (4a),
1-11810 (5a), T11818 (Ca), M1815 (76), &c.*

These low values we were able to associate in most cases with the presence of acid,
but if acid was responsible for the low values the quantity required to lower the
electrochemical equivalent by one part in 10,000 appeared to be very small and could
not easily be detected by litmus and other indicators. The commercial sample W,
had admittedly been crystallised from an acid solution and showed a trace of acid
when carefully tested with blue litmus paper. Finally it was found that the porous
pots which had been cleaned with nitric acid but soaked in water for a fortnight until
all the acid had apparently been removed, gave up acid to the solutions when
a current was passed through them, and this in sufficient quantity to be detected by
litmus. It was thought, however, to be desirable to make direct experiments on the
influence of acids.

Our own experiments have shown that the addition of nitric acid to silver nitrate
solutions produces very irregular results. The normal effect appears to be a small
decrease in the electrochemical equivalent, but this never exceeds about two parts in
10,000, and there is very little difference in the effect produced by a mere trace of
acid which can only be detected with difficulty by means of litmus and that produced
by the addition of acid corresponding to 1 per cent, of the silver nitrate present in the
solution. On the other hand we have found that addition of diluted nitric acid may
produce an increase in the electrochemical equivalent ; this we attribute to the
presence in the nitric acid of oxides of nitrogen or other substances which, like the
hyponitrite, may raise the electrochemical equivalent to such an extent as to mask
completely the small decrease due to the acid.

The irregular effects obtained may be seen from the following summaryt :

I. Normal solution ril830(65b). II. Normal solution 1-11826(666).

0-1 per cent. HNO8 1-11841 (G5c). O'l per cent. HNO, 1-11860 (66c).

1-0 per cent. HNO, 1*11836 (65d). 1*0 per cent. HNO, 1 '11840 (66ci).

III. Normal solution T11830 (46c). IV. Normal solution 1 '11827.

0-2 per cent. HNO, 1-11812 (736). I'O per cent. HNO, M1814( 58c).

1-0 per cent. HNO, 1-11814 (73c). 1-0 per cent. HNO, 1-11810 (75c).

Ilia. 1-0 per cent. HNO, 1-11829 (73d). TO per cent. HNO, 1-11819 (7Sl>).

0-1 per cent. HNO, 1-1 1822 (78c).

* There was, however, a considerable quantity of loose silver in most of the early determinations,
t One scries, No. 53, has been omitted, as the unacidified solution (53c) gave a high value and the nitrate
was therefore not pun1.

4 o 2

596 MR. F. E. SMITH, MR. T. MATHER, AND DR. T. M. LOWRY

In all cases but IV. the nitric acid was purified by distillation from silver nitrate ; in
series III. and Ilia, the acid was added in a concentrated form, in the others it was
diluted and titrated. LEDUC found a diminution in the mass of the deposits of
2 parts in 10,000 when free acid was present.

We conclude, therefore, that whilst the abnormally low values which are observed
from time to time can only be explained by the presence of acid, it may be very difficult
in practice to add nitric acid without at the same time introducing other impurities
which may more than counterbalance the effects produced by the acid itself.

L. Effect of Heating Silver Nitrate.

It has been shown above that the mere act of electrolysis does not cause any
increase in the electrochemical equivalent of a silver nitrate solution, and that in our
experiments no contamination appears to be produced by the changes which take
place at the anode. For a considerable time we were of opinion that atmospheric
contamination with sulphide might in some way produce an alteration in the electrical
behaviour of the solution, but finally we were unable to uphold this explanation of the
heavy deposits. We believe, however, that a clue to the origin of the abnormal
deposits may be found in the behaviour of the nitrate when heated, and, on the other
hand, in the action of the nitrate on the filter paper, to which we have referred above,
but which in our own experiments we have only been able to detect at high
temperatures.

Very early in the course of the investigation we noticed that the fusion of the
nitrate caused an increase ill the weight of the deposit. A perfectly normal salt (H^)
was fused, dissolved in water, and its electrochemical equivalent was found to be
I'll 838 (24c), an increase of 1 part in 10,000. A second similar experiment with a
slightly acid salt (Wt) showed an increase in the electrochemical equivalent from
T11819 (mean of several) to Til 835 (25c), the latter value being again 1 part in
10,000 higher than the normal. It was further noticed that the mother-liquors which
gave such high deposits were all obtained from samples of nitrate which had been
fused or strongly heated at some stage of their treatment.

In order to test the effect of heat alone on the nitrate a q\iantity of the purified
salt was heated to incipient fusion for several hours, and the greater part of the
nitrate was removed by crystallisation. The colourless mother-liquor was electrolysed,
and gave for the electrochemical equivalent the value of T11972 (88c). We therefore
feel justified in suggesting that whilst other causes (action of light, action of filter
paper, &c.) may contribute to the production of heavy deposits, the heating of the
nitrate appears to be one of the most effective ways of producing this effect. It is
possible that traces of hyponitrite may be formed both by the action of heat and by
the reducing action of filter paper, but we do not wish to commit ourselves to the
view that the hyponitrite is the only, or even the main, source of the disturbances

ON THE SILVER VOLTAMETER. 597

which have been noticed by ourselves and others. We hope at some future date to
enquire more closely into the nature of the impurities which affect so strongly the
weight of the deposit from the silver nitrate mother-liquors.

M. Other Electrolytes.

(1) Silver Acetate. — Lord RAYLEIOH* found that the addition of a small quantity
of silver acetate to a solution greatly improved the texture of the deposit, but that
the mass of silver deposited per coulomb was considerably increased thereby. We
prepared a saturated solution of silver acetate and electrolysed in the usual manner ;
the deposit was of very fine texture, and the resulting electrochemical equivalent was
1 '12154 (31c) or 0'3 per cent, higher than the normal. There was evidence, however,
that the deposit was not silver alone, for on stripping a portion of it from the platinum
bowl a yellowish-white film was plainly visible on both silver and platinum where
they had been in contact, t

(2) Silver Chlorate. — Lord RAYLKIGH* also employed silver chlorate as an electro-
lyte, independent of the nitrate, and obtained very satisfactory results from it, and if
the mean value of the electrochemical equivalent is deduced from the chlorate
observations alone, it is higher than the value obtained with the nitrate solutions by
only 6 parts in 100,000. In our first attempt to use silver chlorate as an electrolyte
we used a 10 per cent, solution and obtained 1*11839 (49a) for the electrochemical
equivalent. It was apparent, however, that the electrolyte also contained silver
chloride, for a white precipitate had to be filtered from the original solution, and the
effect of silver chloride in solutions of silver chlorate is possibly the same as in
nitrate solutions. The chlorate was recrystallised to free from chloride and a 5 per
cent, solution used to minimise the effect of any remaining impurity. The resulting
deposit had a matt surface and its mass was 2 parts in 100,000 greater than that from
a 3 per cent, solution of the nitrate, but less by 1 part in 100,000 than that from a 15
per cent, solution (8 la, b, c). The electrochemical equivalent may therefore l>e taken
as I'll 827, and confirms Lord RAYLEIGH'S view that a solution of silver chlorate
gives the same mass per coulomb as one of silver nitrate.

(3) Silver Perchlorate. — CARHART, WILLARD and HENDERSON}: have suggested
the use of silver perchlorate as an electrolyte. They found the deposits from such a
solution to be striated and firmly attached to the bowl, but heavier than the deposits
from the nitrate by about 0'007 per cent. It appears, however, that silver chloride
may also have been present in the perchlorate, as instructions are given by them for
this to be filtered out. A small quantity of perchlorate was prepared for us by some

* RAYF.EIQH and SIDOWICK, 'Phil. Trans.,' 175, p. 411, 1884.

t VAN DIJK found n diminution in the mass of a silver deposit from an acetate solution when it wa«
heated to a high temperature in an electric oven.

I CARHART, WILLARD and HENDERSON, ' Amer. Electrochem. Soc.,' 1906.

:><H MR. F. E. SMITH, MR. T. MATHKK, AND DR. T. M. LOWRY

manufacturing chemists, but it was far from pure, and our results are not therefore
comparable with those of CARHART. The chloride was filtered out, and when a
5 per cent, solution was used, the value T11860 (Bid) milligrammes per coulomb
resulted; subsequently we used a 10 per cent, solution and obtained I'll 840 (836).
We infer that pure perchlorate of silver will give the same value as pure nitrate, but
it is much more difficult to prepare.

(4) Acting on a suggestion of Mr. W. C. D. WHETHAM, we prepared an electrolyte
by dissolving silver nitrate in pyridine. Considerable heat was evolved during
dissolution, but the liquid was quite clear. On electrolysis the deposited silver was
of a brownish-red tint, but when washed with very hot water the intensity of the
colour was considerably reduced. It was apparent, however, that the mass was not
pure silver, and the result (1'11890) (676) is not of very great interest.

(5) Fused Silver Nitrate. — MERRILL* was successful in obtaining coherent deposits
from fused silver nitrate, and compared them with deposits obtained from solutions of
the same salt. He concluded that the masses were identical. We have on several
occasions deposited about 7 grammes of silver on platinum bowls and obtained very
coherent deposits, but errors introduced in the manipulation have so far prevented
us from making a satisfactory comparison with deposits obtained in the usual way.

Summary.

1. It is possible to prepare again and again samples of silver nitrate which give in
the voltameters described in Part I. of this communication values for the electro-
chemical equivalent which do not vary by more than 3 parts in 100,000 on either
side of the mean figure.

2. A standard method of purification is described. Commercial samples are usually
pure, but cannot be absolutely relied on.

3. High values are obtained for the electrochemical equivalent if the solution
contains oxide, carbonate, chloride, nitrite or hyponitrite. Low values are caused by
the presence of acid.

4. Impurities which increase the mass of the deposit per coulomb are usually
substances which are insoluble in water, but soluble in silver nitrate solutions ; they
are therefore precipitated from the impoverished solution at the kathode.

5. Silver chlorate and silver perchlorate appear to give normal deposits, but are
more troublesome in use and have no advantage over the nitrate.

6. There may be slight changes in the electrolyte due to its interaction with filter
paper, but the mass of the deposit is not seriously affected thereby in our size of
voltameter in the course of one electrolysis at ordinary temperatures. It is inadvisable,
however, in measurements of precision, to use an electrolyte more than once.

* MERRILL, 'Phys. Kev.,' 10, p. 170, 1900.

ON THE SILVER VOLTAMKTKK. 599

We desire to thank the Committee of the British Association for grants of money
fin- the purchase of materials; Messrs. JOHNSON, MATTHEY and Co. for the loan of
several plat inum vessels; Mr. L. OEBTLI.NG and Dr. SCOTT for the loan of sensitive
1 i.i lances; and Professor G. VAN DIJK for the loan of apparatus used by him in his
researches. Our thanks are also due to Lord UAYLKIOH and Dr. GLAZEBROOK for
much advice during the progress of the investigation.

[ 601 ]

INUP:X

TO I UK

PHILOSOPHICAL TRANSACTIONS.

SI.IMI> A. V.M.. 207.

A.

AYBTOS (W. E ), MATIIKK (T.), and SMITH (F. E.). A New Current Weigher and a Determination of the Electromotive
Kuroe of tlie Normal Weston Cadiuiuin Cell, 463.

B.

BIOKKB (L.). The Distribution of Blue- Violet Light in the Solar Corona on Auguit 30, 1005, as derived from Photographs
taken at Kalaa-ea-S«nam, Tunisia, 907.

C.

Cadmium cell, the normal Weston (SMITH), 393.

Cordite, modified, investigation of the law of burning of (MANRELL), 243.
Corona, distribution of blue-violet light in, August 3O, 1005 (Bicxin), 307.
Current weigher, a new (ATRTOH, MATIIKB, and SMITH), 463.

CPTIIIIIBTKON (0.) and MBTCALFI (B. P.). On the Refractive Indices of Gaseous Potassium, Zino, Cadmium, Mercury,
Arsenic, Selenium, and Tellurium, 135.

E.
Earth, gravitational stability of the (Lori), 171.

Electric furnace reaction* under high gaseous pressures (IIuTTOV and PBTAYIL), 421.
Electricity, discharge of negative, from hot calcium and lime (HoRTO*), 146.

F.
FILON (L. N. G.). On the Dispersion in Artificial Double Refraction, 263.

O.

Gravitational stability of the earth (Lore), 171.

H.

HOBTOK (FBAKK). On the Discharge of Negative Electricity from Hot Calcium and from Lime, 140.
HUT-TON (E. S.) and PBTATIL (J. E.). Electric Furnaoe Reactions under High Gaseous Pressures, 421.

VOL. CCVIJ, — A 426. 4 H 29.2.08

602 INDEX.

I.

Tonisation produced by hot platinum in different gaies (RicHABDSOX), 1.

J.

Jet-vibration method of investigating surface-tension (PBUKUSEN), 341.

L.

LOVE (A. E. H.). The Gravitational Stability of the Earth, 171.
LOWBT (T. M.). See SM:TII (F. E.), MATHER, and LOWRY.

MACMAHON (Major P. A.). Second Memoir on the Compositions of Numbers, 65.
MANSELL (Major J. H.). Investigation of the Law of Burning- of Modified Cordite, 243.
MATHER (T.). See AYBTON, MATHER, and SMITH ; and SMITH, MATHUU, and LOWBY.
METCALFE (E. P.). See CDTHBEBTSOX and METCALFB.

Numbers, second memoir on the compositions of (MxcMAUON), 65.

P.

PEDEBSBN (P. O.). On the Surface-tension of Liquids Investigated by the Method of Jet Vibration, 341.
PETAVEL (J. E.). See HOTTO.V and PETAVKL.

R.

Refraction, on the dispersion in artificial double (FlLON), 263.

Refractive indices of gaseous potassium, zinc, cadmium, Ac. (CuTHBEBTSou and METCALFE), 135.

RICHABDSOX (O. W.). The lonisation produced by Hot Platinum in Different Gases, 1. •

SMITH (F. E.). The Normal Weston Cadmium Cell, 393.

MATHER (T.) and LOWBT (T. M.). The Silver Voltameter, 545.

See also AYBTON, MATHEK, and SMITH.

Surface-tension of liquids investigated by the method of jet vibration (PEDERBEN), 341.

V.

Voltameter, the silver (SMITH, MATHER, and Lowuv), 545.

W.
Weston cadmium cell, normal (SMITH), 393 ; determination of electromotive force of (AYBTON, MATIIKB, and SMITH), 463.

\

HABBISOM AKD 6ON8, PBINTBB8 IN OBCINABT TO HI* MAJK8TY, ST. MARTIN'S LANE, LONDON, W.C.

/. /•' /•.

Phil. Tmn*. A, wf. 207, /V. 1.

NINE AUTOMATIC EXPOSURES ON ONE PLATE.

0-8 MC.
77

08 MC

55

0-8 M
39

08 MC

27

0-8 MC.
19-7

11-4 MI
187

89 5 M.

16 7

43 MC

187

^Figures in flr»t line = exposure; in second line = ratio of focal-length and equivalent diameter of lens.)

r.

I'tiil. Trans., A, vol. 207, l'l»t< i.

ana

I'll II. Tr«n*., A, rot. 207, Hate 3.

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MICROPHOTOGRAPHS OF CRYSTALS OF MERCUROUS SULPHATE.

Magnification - 25O.

igi. I, 5, 8. KieimreUr'it mttbod of |ira|»r«tHin.

„ «, S. 4. Cbemiml uwthnl of prapamtioo.

„ 8, 7. Kleutnilytio method of |tre|mntu>n.

», 10. Puniioc »ulphnric «riil mctho<l of |>re|«raliou.

,. II. I'urchaiKl from Kmblteum.

U. S. linn,,,, „„•! ./. A'. /'

Phil. Trans., A, vol. 207, Plate 6.

Fig. 1. Large high-pressure furnace (vertical position).

Fig. 2. Large high-pressure furnace (horizontal position).

Ayr! ox, Mather and Smith.

I'l.il. Trans., A, vol. 207, Plate 7.

Fig. 2. Complete current weigher (sides of case removed).

Ayrtnn, Mutlit-i- <iml Smith.

Phil Trans., A, TO/. 207, Plate 8.

Fig. 6. General view of physical ba

Fig. la. End view of current weigher.

Sm i tit, Mather and Lowry.

Phil. Trans., A, vol. '207, Plate 9.

ft
Applied 34

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L82 Seric: itheraatical and

v.207 physical sciences

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